A TUTORIAL ON MULTIOBJECTIVE METAHEURISTICS FOR ...

A TUTORIAL ON MULTIOBJECTIVE METAHEURISTICS FOR SCHEDULING AND TIMETABLING J.D. Landa Silva, E.K. Burke Automated Scheduling, Optimisation and Planning Research Group School of Computer Science and IT, Jubilee Campus, University of Nottingham email: {jds,ekb}@cs.nott.ac.uk http://www.asap.cs.nott.ac.uk

ABSTRACT. Many real-world scheduling problems such as machine scheduling, academic timetabling and personnel scheduling are multiobjective by nature. That is, there exist several criteria that must be taken into consideration when evaluating the quality of the proposed solution or schedule. It is common that some or all of these criteria are conflicting, perhaps incommensurable and set by more than one decision-maker. Among these criteria there are for example: length of the schedule, utilisation of resources, satisfaction of people’s preferences and compliance with regulations. Traditionally, these problems have been tackled as singleobjective optimisation problems after combining the multiple criteria into a single scalar value. On the other hand, a number of multiobjective metaheuristics have been proposed over recent years in order to obtain a set of compromise solutions for multiobjective optimisation problems in a single run and without the need of converting the problem to a single-objective one. Most of these techniques have been successfully tested in both benchmark and real-world multiobjective optimisation problems. However, after reviewing the specialised literature, it is noted that the number of reported applications of these modern multiobjective metaheuristics to the class of scheduling problems is still scarce. This paper attempts to be an introductory tutorial on the application of multiobjective metaheuristics to the optimisation of some multiple criteria scheduling problems. This tutorial presents an introduction to the fundamental concepts of multiple criteria decision-making and multiobjective optimisation, gives an overview of some multiobjective metaheuristics that have been proposed over recent years, revises some of the multiobjective approaches that have been applied to some multi criteria scheduling problems, tries to identify the strategies that have been successful in this area and also seeks to identify some promising research directions that may be benefit a wider application of multiobjective metaheuristics to the optimisation of multiple criteria scheduling problems.

1 Introduction Scheduling can be seen as the arrangement of entities (people, tasks, vehicles, lectures, exams, meetings, etc.) into a pattern in space-time in such a way that constraints are satisfied and certain goals are achieved [Wren1996]. Constructing a schedule is the problem in which time, space and other available resources have to be considered in the arrangement. The constraints are relationships among the entities or between the entities and the patterns that limit the possible arrangements of the schedule. In practice, the scheduling activity can be regarded as a search problem where it is required to find any feasible schedule or as an optimisation problem where the best feasible schedule is sought. In real-world problems, expressing the conditions that make a schedule more preferable than another and incorporating this information into an automated system is not an easy task. In addition, the combinatorial nature of these problems implies exploring huge search spaces and human intervention is often necessary to bias the search towards promising regions. The class of scheduling problems includes a variety of problems such as production scheduling, events scheduling, personnel scheduling and others. Many real world scheduling problems are multiobjective by nature, i.e. it is common that in these problems several objectives exist such as minimising the length of the schedule, optimising the available resources, satisfying the preferences of human resources (personnel scheduling), minimising the tardiness of orders (production scheduling), maximising the compliance with regulations (academic timetabling), etc. Over the years, there have been several approaches used to deal with the various objectives in these problems. Traditionally, the most common way has been to combine the multiple objectives into a single scalar value by using

weighted aggregating functions according to the preferences set by the decision-makers and then to find a compromise solution that reflects these preferences. However, in many real scenarios involving multiobjective scheduling problems, it is preferable to present various compromise solutions to the decision-makers so that the most adequate schedule can be chosen. Although this can be achieved by performing the search several times using different preferences in each time, another approach is to generate the set of compromise solutions in a single execution of the algorithm. This has increased the interest for investigating the application of Pareto optimisation techniques to multiobjective scheduling problems. The aim in Pareto optimisation is to find a set of compromise solutions that represent a good approximation to the Pareto optimal front. In recent years, the number of algorithms proposed for Pareto optimisation has increased tremendously mainly because multiobjective optimisation problems exist is almost any domain. Many metaheuristic approaches that were originally applied to single-objective optimisation problems have been extended to produce multiobjective optimisation variants. Among these, multiobjective evolutionary algorithms have received particular attention since several researchers argue that these methods are particularly well suited to deal with multiobjective optimisation problems. Also, some multiobjective metaheuristics based on local search methods such as simulated annealing and tabu search have been proposed recently. In the context of single-objective combinatorial optimisation problems and in particular scheduling problems, it is usually the case that local search is incorporated into evolutionary algorithms in order to improve the results obtained with these methods. This appears to be true also in the multiobjective case given the evidence reported by researchers in the field. Although there are a considerable number of proposed algorithms for Pareto optimisation, the number of reported applications of these techniques to multiobjective scheduling problems is still scarce. This is particularly true for event scheduling (timetabling) and personnel scheduling (rostering) problems for which the majority of the recent publications still consider the use of aggregating functions to combine the multiple criteria into a single value. This paper attempts to be an introductory tutorial on multiobjective metaheuristics for the optimisation of multiple criteria scheduling problems. The specific aims of this tutorial are: first, to identify the strategies that have been successful in the multiobjective optimisation (using metaheuristics) of some multiple criteria scheduling problems and second, to identify promising research directions that may be interesting to explore in order to strengthen the application of modern multiobjective metaheuristics to these and related problems. The tutorial is organised as follows. The next section presents a brief review of the concepts in multiple criteria decision-making and multiobjective optimisation. That section also provides with a brief description of some of the recently proposed multiobjective metaheuristics and attempts to summarise the key similarities and differences between the various algorithms. Section 3 gives a brief introduction to machine scheduling problems and discusses some issues in relation to real-world scheduling. Then, section 4 describes some multiobjective metaheuristics that have been proposed for multiobjective machine scheduling problems. Description of academic timetabling problems and discussion of their multiobjective nature are presented in section 5. Then, section 6 describes some of the metaheuristic approaches that have been used to tackle academic timetabling problems from a multiobjective perspective. It is noted that although in academic timetabling problems several criteria exist, the application of multiobjective metaheuristics in this context is very rare. Therefore, section 7 presents a brief review of the (singleobjective) metaheuristic techniques that have been used to tackle these problems over recent years. By presenting this review, the present tutorial attempts to contribute to create a bridge between automated timetabling and modern multiobjective metaheuristics as a basis for more research in this area. Section 8 touches on some applications of multiobjective metaheuristics to personnel scheduling problems but no attempt to present descriptions of these problems or detailed review of techniques to tackle them is made. Section 9 discusses some other ideas that have been put forward by researchers in the field of multiobjective optimisation that may be beneficial for strengthening the success of these techniques on scheduling problems. Finally, remarks and some proposed research directions are presented in the last section.

2 Multiple Criteria Decision-Making and Multiobjective Optimisation In combinatorial optimisation problems the aim is to obtain the optimal arrangement of a group of discrete entities in such a way that the additional requirements and constraints are satisfied. If the combinatorial optimisation problem is a multiobjective one, it means that various criteria exist to evaluate the quality of the given arrangement or solution and there is an objective (minimisation or maximisation) attached to each of these criteria. It is commonly the case that some of the criteria are in conflict, i.e. an improvement in one of them can only be achieved at the expense of worsening another. Moreover, some of the criteria may be incommensurable, i.e. the units used to measure the compliance with each of the criteria are not comparable at all. For example, for an examination timetable two of the criteria that may be used to express the quality of the schedule are its length and the satisfaction of students’

preferences. The objectives would be to produce the shortest schedule possible and to satisfy most of the requests from students respectively. These objectives are conflicting because a shorter schedule would permit to have more time for complete marking and administrative procedures but students would prefer to have the longest time possible between exams for preparing them and this would imply a longer schedule. The length of the schedule is expressed in number of timeslots but this metric would be inappropriate to indicate the level of compliance with the preferences of students. To express the degree at which the schedule satisfies the students’ requests, other aspects such as the spread and balance of the schedule and the location of difficult exams within the schedule would be more adequate.

2.1 Search and Decision-Making The first decision that has to be made when dealing with a multiobjective optimisation problem is on how to combine the search and the decision-making processes. This can be done in one of three ways [Steuer1986]: Decision-making and then search. This is also known as the a priori approach because the preferences for each of the objectives have to be set by the decision-makers and then one or various solutions satisfying these preferences have to be found. Search and then decision-making. This is also known as the a posteriori approach because various solutions have to be found and then the decision-makers select the most adequate. All the solutions presented to the decision-makers should normally represent a trade-off between the various objectives. Interactive search and decision-making. In this approach the decision- makers intervene during the search in order to guide it towards promising solutions by adjusting the preferences in the process. Another important decision in multiobjective optimisation is how to evaluate the quality of solutions since the conflicting and incommensurable nature of some of the criteria makes this process more complicated. Also here there are several alternatives [Coello Coello et.al.2002]: Combine the objectives. This is one of the classical methods to evaluate the solution fitness in multiobjective optimisation. It refers to converting the multiobjective problem into a single-objective one by combining the various criteria into a single scalar value. The most common way of doing this is by setting weights to each criteria and add them all together using an aggregating function. Alternating the objectives. This is also an approach that has been used for many years. It refers to optimising one criterion at a time while imposing constraints on the others. The difficulty here is on how to establish the ordering in which the criteria should be optimised since this can have an effect on the success of the search. Pareto-based evaluation. In this approach, a vector containing all the objective values represents the solution fitness and the concept of dominance is used to establish preference between solutions. A solution x is said to be noninferior or non-dominated if there is no other solution that is better that x in all the criteria. Formally, the dominane relation is described as follows: Suppose two distinct vectors V = ( v1, v2,…, vk ) and U = ( u1, u2,…, uk ) containing the objective values of two solutions for a k-objective minimisation problem, then
V strictly dominates U if vi < ui , for i = 1,2,..,k.
V loosely dominates U if vi ≤ ui , for i = 1,2,..,k and vi < ui , for at least one i.
V and U are incomparable if neither V (strictly or loosely) dominates U nor U (strictly or loosely) dominates V. Minimisation is considered here mainly because most of the scheduling problems are of this type (minimise processing time, minimise soft constraints violation, minimise schedule length, etc.) but the above definition is altered in the obvious way for the case of maximisation problems. It is important to note that using strict or loose dominance can have an effect on how the search is performed. This is because if a solution is strictly dominated it means that it is outperformed by the other solution in all criteria while if the solution is loosely dominated it means

that it is outperformed in some of the criteria but as good as the other solution in at least one of them. Then, finding a new solution that strictly dominates the current one may be more difficult than finding a solution that loosely dominates it. This is particularly true in some combinatorial problems in which the connectedness of the search space is such that some solutions are more difficult to reach from the existing one and by using loose dominance more solutions can be the reached.

2.2 Pareto Optimisation When a set of compromise (non-dominated) solutions is aimed (search and then decision-making), these solutions should represent a good approximation to the Pareto optimal front. The Pareto optimal front is the set of all nondominated solutions in the multiobjective space. However, when the solutions in the obtained set do not lie on the Pareto optimal front, one should refer to that set as the obtained non-dominated front or the known Pareto front. Pareto optimisation refers to finding the Pareto optimal front or a set that represents a good approximation to that front and multiobjective optimisation methods usually refer to techniques for Pareto optimisation. The appealing of Pareto optimisation comes from the fact that in most multiobjective optimisation problems there is no such singlebest solution or global optima and it is also very difficult to establish preferences among the criteria previous to the search. Even when this is possible, it may be that these preferences change and then having a set of solutions eases the decision-making process. One of the conditions that must be satisfied for a problem to be considered multiobjective is that the criteria are in conflict. Two objectives are in conflict if the complete satisfaction of one of them prevents the complete satisfaction of the other. If any improvement in one of the objectives induces a detriment on the other, then the objectives are said to be strictly conflicting [Bagchi1999]. It has been said that even if the conflicting nature of the criteria is not proved, Pareto-based metaheuristics would be able to find the ideal solution that is the best in all criteria [Fonseca&Flemming1995]. Since in Pareto optimisation the final outcome must be a set of non-dominated solutions, another important aspect to consider is how to evaluate the quality of the obtained non-dominated front. This is a multiple criteria problem on its own because several aspects have to be considered to determine how good the obtained front is. Among these aspects there are:
The number of non-dominated solutions obtained.
The closeness between the obtained front and the Pareto optimal front (if known).
The coverage of the front, i.e. the spread and distribution of the non-dominated solutions. Several methods have been proposed to evaluate the quality of the obtained non-dominated front in Pareto optimisation and therefore assess the performance of multiobjective optimisers (see [Knowles&Corne2002]). Since the Pareto optimal front is defined with respect to the objective space, is it common that most of the metrics proposed are also defined with respect to the objective space. One aspect that is frequently overlooked is the diversity of the obtained front with respect to the solution space. In fact, when researchers report on the quality of the obtained nondominated sets very rarely information is provided regarding the diversity of the solutions in the decision space. This is extremely important because although the obtained non-dominated solutions may be well spread and distributed over the front in the objective space, it may be that either the solutions are also structurally different (diverse) or very similar among them. Considering diversity in the solution space when assessing the quality of the obtained front becomes even more important in real-world multiobjective combinatorial optimisation problems because the similarity among solutions directly relates to how different the arrangement of the discrete entities is between the solutions. The decision-makers may require solutions that are: Similar in structure and in objective values. For example schedules that are very similar and although each of them is non-dominated, perhaps the decision-makers are interested in a certain part of the trade-off surface. Similar in structure but very different in objective values. As above, the schedules are very similar but the decision-makers want solutions from all over the trade-off surface. Diverse in structure and in objective values. As in the previous case, solutions from all over the trade-off surface are required but the schedules must be very different structurally.

Diverse in structure but similar in objective values. In this case, perhaps the decision-makers require schedules of certain similar quality with respect to the trade-off of objectives but they want to see solutions that actually represent very different schedules. Large multiobjective combinatorial optimisation problems are particularly difficult to tackle. One reason for this is that the size of the search space grows exponentially as the problem size increases making impracticable the application of exact optimisation algorithms. Approximation methods such as problem specific heuristics and metaheuristics are often used to tackle the large instances of these problems to obtain near-optimal solutions in a reasonable amount of computation time. Also, theoretical understanding of the solution space is lacking and as a consequence, in many problems of this type there is no notion of the localization and shape of the Pareto optimal front [Ulungu&Teghem1994]. Considering the fact that many real-world combinatorial optimisation problems are also highly constrained, the scenario is even more complicated. Timetabling, routing, rostering and in general, scheduling related real-world problems are some examples of combinatorial optimisation problems that many times involve multiple criteria and almost always are highly constrained.

2.3 Brief Survey of Multiobjective Techniques This section provides an overview of some proposed techniques for Pareto optimisation but no attempt is made to present an exhaustive survey of the field. This brief overview is limited to multiobjective metaheuristics, in particular to evolutionary algorithms and other approaches based on local search. For a review of other (classical) techniques for multiobjective optimisation refer to [Goicoechea et.al.1982] [Miettinen2001] [Steuer1986] while comprehensive surveys on various approaches for multiobjective optimisation including classical and evolutionary methods can be found in [Coello Coello1999] and [Tan et.al.2002]. After describing some of the most recent multiobjective metaheuristics, this section presents some observations regarding the similarities and differences between these approaches with respect to their application to multiobjective combinatorial optimisation problems. In is not in the interest of this paper to provide detailed descriptions of the algorithms mentioned below, instead further references are provided when appropriate.

2.3.1 Multiobjective Evolutionary Algorithms There is no universally accepted definition of evolutionary algorithm, but in the strict sense an evolutionary algorithm handles a population of solutions, evolves this population by means of cooperation (recombination) and self-adaptation (mutation) and uses a coded representation of solutions [Hertz&Klober2000]. A number of multiobjective evolutionary algorithms have been proposed in recent years and the increasing interest on these methods has motivated the extension of evolutionary algorithms originally proposed for single-objective optimisation to multiobjective variants. Some of these multiobjective evolutionary algorithms are briefly described next. Vector Evaluated Genetic Algorithm (VEGA) [Schaffer1985]. This is perhaps the first genetic algorithm in which the concept of dominance was implemented for the evaluation and selection of individuals. In each generation, a group of individuals is selected according to one of the k objectives in the problem until k groups are formed. That is, each group of individuals excels in one of the k criteria. Then the k groups are shuffled together and the genetic operators are applied to produce the new population. Multiobjective Genetic Algorithm (MOGA) [Fonseca&Fleming1993]. In this algorithm each individual is assigned a rank according to the number of individuals in the population by which it is dominated, i.e. all nondominated solutions are assigned rank 1. The fitness is assigned to each individual using an interpolation between the best and the worst rank. A scheme for niche formation is used in which fitness in the objective domain is shared among non-dominated individuals in order to maintain a uniform distribution of individuals over the trade-off surface. The fitness of all individuals in the same rank is averaged and this value is assigned to all of them. A more recent version of this algorithm is described and compared against other methods in [Purshouse&Fleming2001]. Niche Pareto Genetic Algorithm (NPGA) [Horn et.al.1994]. Here, the selection of individuals is carried out using a tournament scheme based on the concept of dominance. The two individuals competing for selection are compared against a subset of the population and the one that is non-dominated (assuming the other is dominated) is selected for

reproduction. If both competitors are dominated or non-dominated, then a sharing scheme based on the size of the niche (equivalence class sharing) is used to break the tie. The improved version of this algorithm, called NPGA-2 is described in [Erickson et.al.2001]. Non-dominated Sorting Genetic Algorithm (NSGA) [Srinivas&Deb1995]. This algorithm also classifies individuals according to dominance in a ranking scheme similar to the one used in [Fonseca&Flemming1985]. However, a dummy fitness value proportional to the population size is determined for each dominance class. Fitness sharing within the same class is also implemented to help maintaining a well-distributed population over the tradeoff front. Once the whole population is classified, a stochastic remainder proportionate selection scheme is used to ensure that the individuals in the first front get more copies for reproduction that the rest of the population. Updated versions of this algorithm incorporating elitism are described in [Deb2001]. Strength Pareto Evolutionary Algorithm (SPEA) [Zitzler&Thiele1999]. This algorithm was proposed as an approach that incorporates several of the desirable features of other multiobjective evolutionary algorithms. The three features common to other approaches and put together here are: the use of dominance to evaluate and select solutions, the use of additional populations to store non-dominated solutions and the use of a niching or clustering scheme. The particular feature in this approach is that the non-dominated individuals in the external population are used to determine the fitness of individuals in the current population and also participate in the selection process for reproduction. In addition, a niche method based on Pareto dominance is proposed which does not require any measure of distance between individuals as in other clustering techniques. The improved version of this technique, called SPEA2 is described in [Zitzler et.al.2001]. Pareto-Archived Evolutionary Strategy (PAES) [Knowles&Corne2000]. This algorithm starts with one randomly initialised solution and then, one candidate solution is generated in each iteration by means of mutations. An external archive (of limited size) is maintained to collect non-dominated solutions. An adaptive grid that divides the objective space is used to evaluate how much crowded the region in which each solution lies is. The candidate solution is discarded if it is dominated by the current solution or any other solution in the external archive. The candidate solution is added to the archive and becomes the current solution if it dominates the current solution. If none of them dominates the other, the decision on which solution becomes the current solution and whether to add or not the candidate solution to the archive is made based on the crowding mechanism. Other variants of this algorithm with population sizes greater that one, were also proposed [Knowles2001]. Other Multiobjective Evolutionary Algorithms. The algorithms above are just a sample of the vast number of methods proposed in the literature in recent years. Other approaches include the multiobjective messy genetic algorithm (MOMGA) I and II [Van Valdhuizen&Lamont2000] and the Pareto converging genetic algorithm (PCGA) [Kumar&Rockett2002]. Another multiobjective genetic algorithm was proposed in [Murata et.al.1996a] but since this algorithm has been specifically applied to machine scheduling problems, a more detailed review of this approach and its subsequent variants is provided later in section 4. In the last two years, many other extensions of evolutionary algorithms for multiobjective optimisation have been proposed. For example, variants of micro-genetic algorithms, cellular genetic algorithms, particle swarm optimisation methods, agent-based algorithms and others can be found in proceedings of recent conferences (EMO 2001, CEC 2002, GECCO 2002, PPSN VII). For a more detailed review of the principles of evolutionary multiobjective optimisation and recent developments in this field refer to [Coello Coello et.al.2002][Deb2001][Van Valdhuizen&Lamont2000].

2.3.2 Alternative Multiobjective Metaheuristics Another class of metaheuristics for Pareto optimisation are those that explicitly use local search or neighbourhood exploration (instead of genetic operators) to drive the search or as an important component of the process (hybrid approaches). As in the case of multiobjective evolutionary algorithms above, several multiobjective metaheuristics using local search have been put forward in the literature. Some of these multiobjective metaheuristics are briefly described below: Simulated Annealing for Multiobjective Optimisation [Serafini1992]. This was perhaps the first extension of simulated annealing for multiobjective optimisation reported in the literature. The proposed idea was to modify the acceptance criteria of candidate solutions in the original algorithm. Various alternative criteria were investigated in

order to increase the probability of accepting non-dominated solutions. A special rule given by the combination of several criteria was proposed in order to concentrate the search almost exclusively on the non-dominated solutions. Multiobjective Tabu Search (MOTS) [Hansen1997]. This algorithm is a population-based extension of the tabu search metaheuristic that uses a set of weights to guide the search towards the Pareto frontier. Each solution maintains its own tabu list and the weights are adjusted in order to keep the solutions away from their neighbours and therefore attempt to cover the whole trade-off surface. Pareto Simulated Annealing (PSA) [Czyzak&Jaskiewicz1998]. This is a population-based extension of simulated annealing proposed for multiobjective combinatorial optimisation problems. The population of solutions explore their neighbourhood similarly to the classical simulated annealing, but weights for each objective are tuned in each iteration in order to assure a tendency to cover the trade-off surface. The weights for each solution are adjusted in order to increase the probability of moving away from its closest neighbourhood in a similar way as in the multiobjective tabu search algorithm in [Hansen1997]. From simulated annealing, this hybrid metaheuristic borrows the idea of neighbourhood search, probabilistic acceptance of candidate solutions and the dependence of this acceptance from a temperature parameter. From genetic algorithms, the approach incorporates the idea of using a sample population of interacting solutions. Multiobjective Simulated Annealing (MOSA) [Ulungu et.al.1999]. This approach is another extension of simulated annealing in which a weighted aggregating function is used to evaluate the fitness of solutions to attempt approximating the various regions of the trade-off surface. The algorithm works with only one current solution but maintains a population with the non-dominated solutions found during the search. Evolutionary Local Search Algorithm (ELSA) [Menczer et.al.2000]. This is an evolutionary algorithm that uses local selection as the main component in order to minimise the interaction between the individuals in the population. The idea behind this approach is that a population of competing individuals can search the space in a parallel fashion. This algorithm does not use recombination and the only operator to generate new solutions is mutation. The authors stressed that the major strengths of this algorithm are its potential to be implemented in parallel and that it maintains the diversity of the population in a way similar to fitness sharing but more efficiently. Memetic Pareto Archived Evolutionary Strategy (M-PAES) [Knowles&Corne2000a]. This is a memetic variant originated from PAES. This memetic algorithm incorporates a population and a crossover operator but uses the same selection mechanism of PAES. Two archives are used, one is the global archive of non-dominated solutions and another serves as the comparison set in the local search phase. The second archive is emptied after each local search and filled again with solutions from the global archive. The authors reported that this memetic version outperformed the original algorithm on test instances of the multiobjective knapsack problem. Genetic Local Search (GLS) [Jaszkiewicz2002]. This algorithm was proposed to generate a set of approximately non-dominated solutions in multiobjective combinatorial optimisation. This algorithm is a hybrid between genetic algorithms and local search in which a weighted aggregating function is generated at random in each iteration. This function is used to select the solutions that will be recombined to form the offspring and to guide the local optimisation of this offspring. Later in section 4, earlier implementations of genetic local search for multiobjective machine scheduling problems are described [Ishibuchi&Murata1998]. Simulated Annealing for Multiobjective Optimisation [Suppapitnarm et.al.2000]. This is another extension of simulated annealing in which one temperature is associated to each objective in the problem. The algorithm uses only one solution and the annealing process adjusts each temperature independently according to the performance of the solution in each criterion during the search. An archive is used to store all the non-dominated solutions visited. Other Multiobjective Metaheuristics Using Local Search. Many other approaches of this class have been proposed and investigated in the literature. For example, the tabu search variant of [Baykasoglu et.al.1999] maintains a single solution but additional lists of non-dominated solutions found during the search are kept in order to seed and guide the search. Another tabu search approach using weights adaptation was proposed specifically for the biobjective knapsack problem in [Gandibleux&Freville2000]. Other multiobjective variants of ant colony optimisation, hybrids between tabu search and evolutionary algorithms and other implementations of multiobjective genetic local search can be found in proceedings of recent conferences (EMO 2001, CEC 2002, GECCO 2002, PPSN VII).

2.4 Some Observations on Multiobjective Metaheuristics The following observations can be made with respect to the various multiobjective metaheuristics proposed in recent years: Almost all multiobjective evolutionary algorithms use Pareto dominance for the selection of individuals during the search while many of the alternative multiobjective metaheuristics (the ones using local search) employ a different method, in most of the cases a weighted aggregating function. It is argued in [Jaszkiewicz2001] that aggregating functions are advantageous because they force the solutions to explore certain areas of the solution space and these functions not necessarily should be interpreted as a transformation of the multiple objectives into a singleobjective, i.e. they are just a tool to guide the search in multiobjective optimisation and it does not imply that they reflect the decision-makers’ preferences. In his opinion, dominance assures only convergence but not dispersion over the front while aggregating functions are able to produce points in all regions of the front. In [Jaszkiewicz2002], the author suggests that Pareto dominance is not well suited for combination with local search. It should be noted that in the opinion of Knowles and Corne, the mutation operator used in their Pareto archived evolutionary strategy can be regarded as a form of local search [Knowles&Corne2000]. This can be argued, but the important point here is that, on one hand this implies that local search can successfully be combined with Pareto dominance and in the other hand, this also means that the utility of local search in multiobjective optimisation is confirmed. This emphasises the debate on the use of weighted aggregating functions or Pareto dominance for multiobjective optimisation. Generally, multiobjective evolutionary algorithms use standard genetic operators and the differences between these algorithms concentrates on the strategies used for selection and diversification. The alternative approaches employ neighbourhood search, which needs to be designed according to the problem. Many multiobjective evolutionary algorithms use diversification mechanisms based on clustering or fitness sharing to disperse the population over the trade-off surface while it is common that the alternative multiobjective metaheuristics employ the variation of weights for this purpose. The majority of papers reporting on the application of techniques to multiobjective combinatorial optimisation problems use local-search based heuristics or metaheuristics. In connection with this, some surveys on metaheuristics for multiobjective combinatorial optimisation problems have been published and it has been noted by their authors that methods based on local search appear to be more frequently applied to this class of problems [Jones et.al.2001][Ulungu&Tegehm1994]. Ehrgott and Gandibleux found in their survey that interactive search (modification of the preferences by the decision-makers) is the most commonly used form of search in practical multiobjective combinatorial optimisation problems [Ehrgott&Gandibleux2000]. On the other hand, most of the multiobjective evolutionary algorithms are tested on continuous functions like the ones proposed in [Deb1999][Deb et.al.2000][Deb et.al.2002]. It is true that tests have been also carried out on the application of multiobjective evolutionary algorithms to some multiobjective combinatorial problems but not at the extent of the alternative multiobjective metaheuristics [Knowles2001][Tan et.al.2002][Zitzler et.al.2000][Zydallis et.al.2001]. Having presented this brief review on the main features of the various multiobjective metaheuristics proposed in the literature, the next sections concentrate on the approaches that have been applied to some types of multiobjective scheduling problems.

3 Machine Scheduling Problems 3.1 Description of Machine Scheduling Problems For the purposes of this tutorial, machine scheduling refers to problems where jobs or tasks have to be scheduled for processing in one or more machines. A variety of machine scheduling problems exist, three of them are: flow shop, job shop and open shop scheduling problems. A brief description of these problems is given next but for further details on the models of these and other machine scheduling problems refer to [Pinedo1995]. In machine scheduling, a job can be seen as a distinct and identifiable task that has to be processed either in one step in one machine or in several steps in different machines. In these problems a finite set of jobs has to be scheduled for processing in a finite set of machines. The following notation is of relevance in machine scheduling problems:


n is the number of jobs or tasks.














m is the number of machines available. (i,j) refers to the processing of job j on machine i. p(i,j) is the time that takes to process job j on machine i. d(j) is the due date which is the committed completion time (deadline) for job j. c(j) is the completion time of job j which refers to the actual time taken for processing the job. e(j) is the earliness of job j, i.e. how much time the job was completed before the due date, e(j)=max(0,d(j)-c(j)). l(j) is the lateness of job j, i.e. the delay on the completion of the job with respect to the due date, l(j)=c(j)-d(j). t(j) is the tardiness of job j, i.e. the time that the job is actually completed late, t(j)=max(0,c(j)-d(j)). The makespan or total completion time is equal to the completion time of the last job. Number of late jobs is the number of jobs for which t(j)>0. Pre-emption means that a given (i,j) can be interrupted and resumed later, i.e. the processing of a job in a machine can be suspended in order to process another job and the resumed later.

Flow Shop Scheduling. In this problem there are n jobs or tasks that have to be processed in each of the m available machines, i.e. each job consists of m steps. The processing of each job is carried out in the same sequencing through the processing stages, i.e. from the first to the last machine. After the processing of the job is finished in machine i the job joins the queue in machine i+1. Then each machine i is used to process the ith step of each job. The problem here is to find the sequence in which the jobs should be processed so that the objectives are achieved. Job Shop Scheduling. This is a more general case of the flow shop scheduling problem in which the sequencing of each job through the machines is not necessarily identical. As in a flow shop, here there are also n jobs consisting of m steps and m machines are available. The sequence of steps within each job are predefined and fixed. Open Shop Scheduling. The open shop is a more general case of the job shop scheduling problem. As before, there are n jobs consisting of m steps to be processed in m available machines. The sequencing of each job through the machines can be different and finding the optimal sequencing for each of the n jobs is part of the problem. Since the sequence of steps within each job has to be determined in addition to the jobs processing schedule, the search space is even larger that in the job shop scheduling problem.

3.2 Some Issues on Real Machine Scheduling Of course elaborated models of machine scheduling problems representing realistic situations exist in the literature together with proposed techniques to tackle them. However, there are several differences that are found in real-world production scheduling problems with respect to the descriptions of the theoretical machine scheduling problems [Pinedo1995]. Some of the differences that are normally encountered between the theoretical models and real machine scheduling problems are as follows. For example, it is frequently assumed that all the jobs to be scheduled are available for processing in the beginning, each machine can process only one operation at a time, no job is processed more than once in any machine, there is enough storage between machines and the times for setting-up the machines are included in the processing time and they are not dependent on the selected sequence. The machine scheduling activity is dynamic in the real world. That is, a number of jobs need to be processed at any time but new jobs are continuously added or some jobs can be suspended (pre-emption) and eliminated from the schedule. Moreover, the priority of the existing jobs may change in real time or the priority of the new jobs may be such that the schedule has to be modified. Another situation that can occur is that the availability of machines and/or resources may change and this again affects the current schedule. It is also common that machine scheduling problems are treated as deterministic, i.e. assuming that the processing times of the jobs is each machine are known and fixed. Stochastic and fuzzy job scheduling problems are also considered in the literature [Bagchi1999][Pinedo1995]. In the first case the processing times follow a probability distribution while in the second case the processing times are specified in a fuzzy way according to expected completion times. Although these non-deterministic variants exist and they might represent real problems more closely, most of the research has been dedicated to the deterministic scheduling problems. This dynamic nature of real world production scheduling problems emphasises the importance of the rescheduling capability of good scheduling systems. The assumptions about the availability and conditions of the

machines are often over simplified in the theoretical models while in real environments, machines may not be available for certain periods of time. Although more than one objective can be considered in the model, the relative importance of these can vary over the time. Machine scheduling is closely related to the available workforce, so the availability of this resource has a direct impact on the execution of the schedule. All the above are reasons that make the application of multiobjective techniques to machine scheduling problems appealing. Using these methods, a set of schedules representing a trade-off among the various criteria can be generated so that other schedules are available and can be considered as alternative solutions (after perhaps small modifications) when the conditions in the environment change.

3.3 Theory and Practice in Machine Scheduling Research Since many scheduling problems are intractable or NP-hard it is common that heuristic techniques are applied to obtain an acceptable or sub-optimal schedule in a reasonable amount of processing time. In fact, due to their recognised importance and difficulty, scheduling problems are among the most investigated optimisation problems and new algorithms are commonly tested in these problems. For the last forty years or so, many heuristics have been proposed for tackling the various variants of job scheduling problems. Heuristic methods include those specialised algorithms that have been designed for a particular scheduling problem and also more general heuristics that can be applied (after some tuning) to a variety of optimisation problems. Reviews of some of the specialised heuristics for job scheduling problems can be found in [Bagchi1999][Pinedo1995]. This tutorial is concerned only with the second class of heuristics known as metaheuristics. Metaheuristic techniques include local search methods, threshold acceptance, tabu search, simulated annealing, variable neighbourhood search, genetic algorithms, neural networks, ant colony optimisation, hyperheuristics and many others. Recently, the workshop Scheduling: Bringing Together Theory and Practice was organised during the 2002 Genetic and Evolutionary Computation Conference (GECCO 2002). The aim of this workshop was to increase the cross-fertilisation of ideas between the academic community and those practitioners solving real-world scheduling problems. Several interesting observations resulted from that meeting:
Almost every type of metaheuristic has been applied to scheduling problems.
Partial schedules often exist in real problems and rescheduling or dynamic scheduling is very often needed.
It appears that it is more difficult to apply genetic algorithms to rescheduling than to create schedules from scratch.
The application of genetic algorithms to real industrial scheduling problems is still an art not a science.
Crossover and mutation operators are designed to exploit the problem knowledge and trial and error is commonly needed.
When implementing a genetic algorithm for a scheduling problem, the design of genetic operators and parameters tuning requires about 30% of the time while the design of the evaluation function requires about 70% of the time.
For a scheduling system being successful, the scheduling algorithms must solve the particular real problem not a simplified abstraction.
Flow shop and job shop have been addressed with many algorithms over forty years but real-world problems are different.
The scheduling system in the real-world must be fast and the algorithm is only a part of developing such a system therefore a fast algorithm does not imply that it will permit the development of a fast system which includes user interfaces and data sources.
Rescheduling is a very important task is real scenarios, therefore algorithms need to support dynamic scheduling.
Computational complexity of the theoretical problems does not mean complexity of the real world problem. Then scheduling researchers should tackle more real-world problems more often.
The only variability on the benchmarks used is commonly only on the numbers. Also benchmark problems are static and do not reflect the dynamic and interaction with humans that exist in real-world problems. But new benchmarks that are more similar to these problems are not easily accepted because existing algorithms find difficult to handle the many different constraints and the dynamics of the scheduling process. As many other metaheuristics, a number of genetic algorithms have been applied to various scheduling problems and many researchers have found that it is essential to incorporate knowledge about the problem domain, constraint-

handling techniques, specialised genetic operators and local search heuristics in order to obtain good results [Ishibuchi et.al.1997][Murata et.al.1996][Varela et.al.2001]. The resulting algorithms are commonly known as hybrid genetic algorithms, memetic algorithms, genetic local search, Lamarckian genetic algorithms and other names. The next section describes some of the multiobjective metaheuristics proposed to tackle some multiobjective machine scheduling problems.

4 Multiobjective Approaches for Machine Scheduling Several objectives can be considered in machine scheduling problems such as the following ones: minimising the makespan or completion time, minimising lateness, minimising tardiness, maximising earliness, matching the due date as closely as possible, maximise the speed at which the jobs flow during their processing (flow time), etc. Most of the reported applications of multiobjective metaheuristics on multiple criteria machine scheduling have considered two or three objectives and many have concentrated of flow shop scheduling problems. A literature survey on multiple criteria scheduling problems up to 1995 is available in [Nagar et.al.1995]. In that survey it was noted that the research in this area was scarce, specially in multiple machine multiple criteria problems like the ones described above, perhaps due to the complexity of these problems. Other observations made were that not much research has been carried out using stochastic models and that most of the approaches considered criteria related to the flow time and the tardiness of jobs combined in weighted aggregating functions. In that survey it was concluded that the most common methodologies used by the time of publication, included branch and bound and dynamic programming and that these techniques were not very successful in efficiently solving large problems. In the following subsections, some of the recent developments (from 1996) on the application of multiobjective metaheuristics to multiple criteria machine scheduling problems are described.

4.1 Metaheuristics Based on Local Search Marett and Wright presented a study on the application of three techniques based on local search to multiobjective flow shop scheduling problems [Marett&Wright1996]. Although they did not aimed Pareto optimisation, it was decided to include their paper in this review because they made some interesting observations regarding the complexity of these problems from the local search point of view. Since, as mentioned before, almost all the proposed multiobjective metaheuristics for scheduling include some form of local search, the results presented by Marett and Wright are of relevance to this tutorial. They assessed the performance of a simple descendent method, a tabu search technique and a simulated annealing algorithm according to the degree of complexity in the multiple criteria flow shop problem. They used test problems with 4, 3, 2 and 1 objectives using the combinations of total setup time, total setup cost, total holding time and total late time but combined them into a single scalar value. One neighbourhood structure was used in all methods: the swap or exchange of two jobs. They observed different performances of the three techniques for different degrees of problem complexity but in general they noted that exploring not all but a subset of neighbours produced much better results, an observation that was also made in [Murata et.al.1996] for single-objective flow shop problems. Marett and Wright proposed two metrics to measure the complexity of a combinatorial optimisation problem: the mean steepest descend length and the first autocorrelation. The first metric is a measure of the number of complete neighbourhoods that have to be examined before a local optima is found while the second metric is based on a random walk of the solution space and then observing the rate of improvements made.

4.2 Multiobjective Genetic Algorithms Murata et.al. proposed a multiobjective genetic algorithm for the flow shop problem with two and three objectives [Murata et.al.1996a]. The criteria considered were: total makespan, total tardiness and total flow time. Their genetic algorithm used weighted vectors generated at random in each selection step. That is, before selection, a vector of weights is generated at random and all the individuals in the population are evaluated using that vector. Then two individuals are selected according to a probability function before applying the genetic operators to produce one offspring. A secondary population of non-dominated solutions is maintained and some individuals of this elite population are copied to the next generation. The randomly generated weights aim to specify different search

directions towards the Pareto optimal front. In addition to the secondary population, elite individuals with respect to each of the k objectives are maintained. The two-point crossover and the shift mutation were used since they observed that these worked fine in their previous work [Murata et.al.1996]. For the bi-objective case, the weights were generated according to (4.1) and the solution fitness calculated using (4.2) as shown below. w1 =

i −1 and N selection − 1

w 2 = 1 − w1 for i = 1, 2 ,..., N selection

f ( x ) = w1 f 1 ( x ) + w 2 f 2 ( x )

(4.1) (4.2)

where Nselection is the number of selection steps in each generation of the algorithm. For the k-objective case, Murata et.al. proposed to generate the weights and calculate the solutions fitness according to (4.3) and (4.4) respectively. rnd i (4.3) w = for i = 1, 2 ,..., k i

k

∑ rnd j j =1

f ( x ) = w1 f 1 ( x ) + ... + w k f k ( x )

(4.4)

where rndi and rndj are non-negative random numbers. Then k weights are generated in each of the Nselection selection steps to choose a pair of parents for recombination. Murata et.al. found that their approach with variable weights was capable of approximating the Pareto optimal set in non-convex fronts and produced better results than the VEGA.

4.3 Extensions to the Multiobjective Genetic Algorithm The multiobjective genetic algorithm described above, was later hybridised with local search in [Ishibuchi et.al.1997] and applied to multiobjective flow shop scheduling problems in [Ishibuchi&Murata1998]. The new version used the strategy of specifying different random search directions for each selection of parents according to (4.3) but now, after each offspring is generated by means of the genetic operators, local search is applied to the new individual in order to improve it. The same mutation operator was employed to explore the neighbourhood in the local search. Also, the same vector of weights generated to select the parents is also used to guide the local search. The number of neighbours explored during the local search was a subset of the whole neighbourhood as suggested in [Murata et.al.1996] as a way of controlling the computation time spent by the local search. The elitist strategy was also slightly modified so that the local search is also applied to some randomly selected individuals from the secondary population of non-dominated solutions. The weighted vector used for the local search of each elite individual is the one of its parents and if no parents exist (an initial generated solution), random weights are used. The authors compared their approach against the VEGA and against a genetic algorithm with fixed weights and found that the proposed algorithm outperformed these two methods. Ishibuchi and Murata also carried out experiments to assess the dependence of the their hybrid algorithm to parameters such as the number of neighbours examined in the local search, the number of non-dominated solutions copied from the secondary population and the multipliers used for the normalisation of objectives. From their results they concluded that certain values were required in order to obtain better results, i.e. the algorithm was sensitive to these parameters. The above multiobjective genetic local search algorithm was extended to a multiobjective cellular genetic local search algorithm by applying a cellular structure and a form of relocation procedure by immigration [Murata et.al.2000]. In a cellular algorithm each individual resides in a cell of a spatially structured space similar to the concept used in the PAES for diversity and niching. A different weight vector is assigned to each cell so that for a kobjective problem the space is structured in a k-objective weight space. Later, Murata et.al. proposed a proportional weight specification method and incorporated it into the multiobjective genetic algorithm and into the cellular variant in order to examine the effect of this new mechanism on the performance of these algorithms in multiobjective flow shop scheduling problems [Murata et.al.2001]. The weights are generated systematically in order to allocate cells of

uniformly distributed weight vectors. The distance between cells is measured with the Manhattan distance, i.e. the distance between two cells with weights w = (w1, w2, …, wn) and v = (v1, v2, …, vn) is given by:

distance ( w , v ) =

k

∑| w

i

− vi |

(4.5)

i =1

and the neighbourhood of a weight vector w is given by neighbourhood(w)={v | distance(w,v) ≤ D}

(4.6)

where D is a predefined distance that is set as a parameter of the algorithm. To generate an individual in a cell, two parents are selected from its neighbourhood. The fitness during the selection of the neighbours is calculated using the weighted vector of the cell to which the individual is being generated. The cellular structure can be seen as restricting the genetic operations to be performed on individuals that are close and not to far away. Murata et.al. applied their algorithm to flow shop scheduling problems with two and three objectives. They compared three algorithms, their new cellular multiobjective genetic algorithm and the multiobjective genetic algorithm with random weights and with weights generated by the new proposed mechanism. They observed that the new weights generating method improved the performance of the algorithms and also found that the level of restriction in the genetic operations (D, the distance for neighbouring solutions) in the cellular approach had an effect on the performance of the algorithm, i.e. it was necessary to tune this parameter. Later, Ishibuchi et.al. modified the multiobjective genetic local search algorithm by selecting only good individuals for applying the local search phase instead of applying it to all the offspring [Ishibuchi et.al.2002]. In this new version of the algorithm, the authors addressed the two difficulties that they found when hybridising genetic algorithms with local search: how to specify the objective function and how to establish the balance between local search and genetic search. In particular, the second aspect becomes more important if the time spent evaluating local search in the problem being tackled is considered expensive. As described above, in the previous versions of the algorithm it was found that not examining all neighbours during the local search was beneficial. In this new version the two modifications proposed were:
Only a small number of good offspring are selected for applying local search and,
The local search direction is specified according to the localization of the solution in the objective space. Basically they modified the step for selecting individuals for local search. As in the previous version of the approach, a random vector of weights is generated and then, using tournament selection with replacement one solution from the population is selected and added to the local search pool. Once this pool is complete, a number NLS of solutions are selected from this set for applying local search. The local search direction of each solution is specified by the weighted vector used in the selection of that solution when constructing the local search pool. The new population of solutions is composed by the improved NLS solutions and the other non-selected solutions in the local search pool. Ishibuchi et.al. compared this version with their previous ones: without local search and applying local search to all offspring and found that the new version outperformed the previous ones. They also compared the new proposed version against the SPEA and the NSGA-II. They observed in their experiments that their modified algorithm was now competitive with these two contemporary algorithms in terms of solutions quality but in terms of computation time efficiency their algorithm was better. They also analysed the effect of the number of solutions selected for local search on the performance of the algorithm and observed that it was necessary to tune this parameter in order to obtain better results. In summary, the authors proposed the specification of an appropriate search direction for the local search by using the tournament selection and the application of local search to only good solutions as additional strategies for establishing a good balance between local search and genetic search. The same investigation described in the above paragraph was presented in [Ishibuchi et.al.2002a] with additional experiments to assess a hybrid version of the SPEA. In that paper, the authors incorporated the same local search components as in their multiobjective genetic local search algorithm to the SPEA and noted a small improvement in the performance of this algorithm. In particular, the improvement observed was in terms of the computation time and the authors suggested that this was because the local search is performed faster than the genetic search. They also examined the effect of the balance in the local search and genetic search within the hybrid SPEA and observed that for the hybrid variant of the SPEA larger values of number of neighbours explored were better. In general they

concluded that the appropriate balance between local search and genetic search depends on two aspects: the algorithm and the available computational time. It can be noted that since the implementation of the multiobjective genetic algorithm proposed in [Murata et.al.1996a], additional strategies have been incorporated to create different versions of the algorithm and improve the results on multiobjective flow shop scheduling problems. It is noted that the suggested modifications range from the adequate selection of genetic operators to fine-tuning the balance between local search and genetic search. In general, in those works it has been shown the importance of local search for the good performance of these algorithms when tackling multiobjective flow shop scheduling problems. Fig. 1 illustrates the various strategies that have been proposed in the various implementations of the multiobjective genetic local search algorithm described above. selected genetic operators and random weighs or cellular structure current population

update

select good solutions and specify good search directions

intermediate population

local search exploring a subset of the neighbours

elite solutions for local search

elite solutions

secondary population of non-dominated solutions

new population

update

secondary population of non-dominated solutions

Fig. 1. The various strategies proposed in the implementations of the multiobjective genetic local search algorithm.

4.4 Hybrid Multiobjective Evolutionary Algorithm A hybrid evolutionary algorithm was proposed for the flow shop scheduling problem with two objectives: the minimisation of makespan and the minimisation of total tardiness in [Talbi et.al.2001]. Their hybrid consisted on applying a genetic algorithm to obtain an approximation to the Pareto front and then applying local search to the obtained front. That is, once a non-dominated front is obtained using the genetic algorithm, the local search explores neighbours of the solutions in this front and updates the set accordingly until no new non-dominated neighbours are found. The neighbourhood exploration was carried out using the mutation operator of the genetic algorithm. The crossover and mutation operators used were those employed in [Murata et.al.1996a]. An interesting aspect of the study presented by Talbi et.al. is that the authors investigated various selection criteria:







Combination of objectives using weights. The parallel selection strategy used in the VEGA. The selection strategy used in the NSGA. A non-dominated sorting selection. A weighted average ranking where individuals are ranked according to the different objectives separately. An elitist method where a population of non-dominated individuals is maintained and it participates in the selection for reproduction.

They observed in their experiments that elitist selection was the most beneficial and that the non-Pareto based selection schemes (combination using weights and weighted average ranking) seemed not suitable to the problem. They also found that tuning the elitism pressure was important since high pressure intensifies exploitation tendency of the good solutions while low elitism pressure favours exploration of new regions in the search space. Another interesting aspect of the study by Talbi et.al. was that they compared three ways of fitness sharing: genotypic sharing, phenotypic sharing and a combined approach. In the decision space (genotypic sharing) the

distance between two individuals x and y was measured according to the distance between the schedules (represented by a permutation) given by: dist1( x , y ) = |{( i , j ) ∈ J x J | i precedes j in the solution x and j precedes i in the solution y }|

(4.7)

In the bi-objective space (phenotypic sharing) the distance between two individuals x and y was given by: dist2( x , y ) = | f1(x) – f1(y) | + | f2(x) – f2(y) |

(4.8)

Finally, the third approach combined the distances in both the decision and objective spaces and the distance between two individuals x and y was given by: sh(x,y) = 1 −

dist 1( x , y ) if γ1

sh(x,y) = 1 −

dist 2 ( x , y ) if γ2

sh(x,y) = 1 −

dist 1( x , y ) dist 2(x,y) if dist 1(x,y) < γ 1, dist 2(x,y) < γ 2 γ1γ 2

dist 1(x,y) < γ 1, dist 2(x,y) ≥ γ 2

(4.9)

dist 1(x,y) ≥ γ 1, dist 2(x,y) < γ 2

where γ1 and γ2 are parameters set to 4.0 and 1.0 respectively. Talbi et.al. observed in their experiments that when their algorithm used phenotypic sharing, it produced closer approximations to the Pareto front but when using genotypic sharing solutions were found in some areas that the other variant did not cover. They decided to use the combined sharing approach in their final implementation because it appeared to outperform the other two methods by helping to obtain closer approximations to the Pareto front also better coverage of this front. In their experiments, they also observed that their hybrid evolutionary algorithm was more beneficial as the problem size increased. They also presented a parallel version of their approach in which the speed was improved since the sequential version was considered slow.

4.5 Dynamic Mutation Pareto Genetic Algorithm Basseur et.al. presented an approach called dynamic mutation Pareto genetic algorithm and applied it to the flow shop scheduling problem with two objectives: minimisation of total makespan and minimisation of total tardiness [Basseur et.al.2002]. The distinctive feature of their algorithm is that it uses different genetic operators in a simultaneous and adaptive manner during the search. In their approach, several mutation operators are given the same probability at the beginning of the search and then they are chosen dynamically during the search. The individuals are evaluated before and after the application of the mutation operators. Then, for each mutation operator an average growth value is calculated and used to adjust the probability assigned to each mutation operator. After applying a mutation operator M, a solution M(x) is generated from a solution x. The progress of a mutation operator M applied to a solution x is 1 if the solution x is dominated by M(x), 0 if x dominates M(x) and 0.5 otherwise. Then, the average Progress(M(i)) is calculated by summing all the progresses of the mutation operator M and divided by the number of solutions to which the mutation operator was applied. The probability of each mutation operator is adjusted by: PM (i ) =

Progress( M (i )) n

∑ Progress( M ( j ))

∗ (1 − n ∗ δ ) + δ

(4.10)

j =1

where n is the number of mutation operators and δ indicates the minimal ratio value of each operator. In their implementation, Basseur et.al. used two mutation operators: an exchange (swap) between jobs and the insertion operator, which is the same as the shift change operator, used in [Murata et.al.1996a]. They used fitness sharing with a combination of the distance in the solution and decision spaces [Talbi et.al.2001]. Their hybrid consisted in a genetic algorithm followed by a memetic algorithm applied only during a few generations due to its

more expensive computational cost. They found improvement over previous results [Talbi et.al.2001] in both the proximity to the Pareto front and the diversity of the solutions found.

4.6 Semi-Exact Population Heuristic Gandibleux et.al. proposed the idea of first generating the set of supported (non-dominated solutions produced using weighted vectors) solutions using an exact or heuristic method and then use these solutions to improve the front by using a population heuristic [Gandibleux et.al.2001]. The supported solutions are considered as good genetic information available and used by a population heuristic. The set of supported solutions help to achieve a faster convergence to the Pareto front and also to maintain diversity of the population. They applied their concept to two bi-criteria combinatorial optimisation problems. One was the single machine scheduling problem namely permutation scheduling with two objectives: the minimisation of the total flow time and the minimisation of the maximum tardiness. The other problem was the bi-ojective knapsack problem. The main characteristics of the population heuristic that they used in the second phase of their approach are:
All solutions ranked one with the non-domination ranking mechanism are copied to the next generation.
During selection, some good solutions with respect to each objective are copied to the new population as in the VEGA.
Among the solutions not selected as above, tournament selection is applied based in dominance with sharing.
In the initial population, besides the solutions generated randomly, some good solutions with respect to each objective are computed and added to the initial population.
Local search is applied to all elite individuals except to those that already received local search in the previous generations. Gandibleux et.al. observed in their experiments that seeding elite solutions permitted to propagate the superior genetic information to other individuals during the evolution process. Also when all supported solutions were used to seed the search, a big impact was observed since the computation time and the number of generations needed was reduced considerably. They suggested that this two-phase method or semi-exact approach can be specially useful in problems for which efficient method exist to solve the single-objective version of the problem or for problems for which an efficient greedy algorithm exist. In their experiments, they observed that their approach showed fast convergence to the front, good quality in term of the closeness to the front and uniformity of the solutions along the front.

4.7 Implementations of the Non-Dominated Sorting Genetic Algorithm Bagchi applied the original NSGA and also an extension of that algorithm to multiobjective flow shop, job shop and open shop scheduling problems in [Bagchi1999][Bagchi2001]. The extended approach described by Bagchi, called elitist non-dominated sorting genetic algorithm (ENGA), was an elitist version of the original algorithm in which the selection mechanism was modified to consider the combined parents and the offspring to form the next generation. The author observed that non-dominated sorting mechanism augmented with elitism was capable of improving the speed of convergence towards Pareto optimal solutions. Brizuela et.al. also applied the NSGA to the flow shop scheduling problem with three objectives: minimisation of the makespan, minimisation of the mean flow time and minimisation of mean tardiness [Brizuela et.al.2001]. They studied the effect of the genetic operators used on the dominance properties of the solutions generated. Three different crossover operators and three different mutation operators were implemented and compared. The aim of their study was to identify the relationships between the operators used and its success on finding non-dominated solutions. They expressed that since in continuous optimisation one Pareto neighbourhood is close to each other but in combinatorial optimisation this property does not hold, it is interesting to know which operator better goes from one Pareto solution to another. Then, they compared the various mutation operators and observed an influence of the operator used on the quality of the non-dominated solutions generated, effect that can be translated into a concept of non-dominated local search. A second set of experiments was carried out using the various crossover operators in order to study the relation between the distance between parents in the decision space and the dominance relation between the parents and the offspring after the crossover. They observed that a combination of the genetic operators

used in [Murata et.al.1996a] performed the best. They also used two distance measures, one in the solution or decision space and the other in the objective space. For measuring the distance between two solutions x and y in the decision space, a matrix n x n is associated to each permutation of the n jobs representing a schedule. Each element of the matrix aij = 1 if job j is scheduled before job i and zero otherwise. Then the normalised domain distance between two individuals is give by: n

dn( x, y ) =

n

∑∑ aij ( x) ⊕ aij ( y) j =1 i =1

n( n − 1)

(4.11)

where ⊕ represents the exclusive-or logical operation and n(n-1) is the maximum number of different elements between two given associated matrices. The Euclidean distance was used for measuring the difference between individuals in the objective space. So the objective function distance between two solutions x and y with k objectives is given by: ofd ( x, y ) =

k

∑ ( f j ( x) − f j ( y))2

(4.12)

j =1

Brizuela et.al. applied the selected operators to the NSGA and outperformed the results obtained by the modified version of [Bagchi 1999]. They expressed that their experiments offered an insight on how non-dominated local search can be preformed since different operators produce different results with respect to non-dominance and this may be a first step on the analysis of the landscape in multiobjective combinatorial optimisation problems.

5 Timetabling Problems 5.1 The Timetabling Activity Timetabling is the activity of scheduling a set of meetings or events in terms of time and occasionally in terms of space in such a way that certain requirements and constraints are satisfied. In timetabling, the allocation of resources other than people and locations for the meetings is usually not part of the problem. In many timetabling problems the meetings to be scheduled are already specified and then the problem is to schedule them into the available timeslots and locations. However, in some timetabling problems the creation of meetings (relationships between the entities such as teacher-class or exam-invigilator) is also part of the timetabling activity. Constraints can be hard or soft. Hard constraints are also called first order or primary constraints and they must be satisfied so that the timetable can be considered feasible. Soft constraints are also called second order or secondary constraints and they can be violated but this situation is penalised according to their relative importance. The variety and particularity of timetabling problems is one of the reasons that have motivated the interest of many researchers in this specific domain. Timetabling problems include academic timetabling, sports timetabling, employee timetabling, conference timetabling and others. Among these, academic timetabling is perhaps one of the most investigated timetabling problems. This section concentrates on academic timetabling. An effective timetabling activity in academic institutions is crucial for the satisfaction of educational requirements and the efficient utilisation of human and space resources [Rankin 1996]. Academic timetabling problems include the school timetabling problem (class-teacher timetabling), the university timetabling problem (course timetabling) and the examination timetabling problem.

5.2 Description of Academic Timetabling Problems The academic timetabling problem has many variants and each problem instance is virtually different. It should be noted that many models and formulations have been proposed to describe academic timetabling problems. This section provides only with a brief description of some academic timetabling problems. For more elaborated models and formulations refer to [Carter et.al.1994][Costa1994][DeWerra1985][Schaerf1999].

School timetabling. In general terms this problem refers to assigning timeslots and locations so that meetings between teachers and classes can take place. The main two differences between this and other academic timetabling problems are: 1) the students are grouped in classes and, 2) the meetings and number of them are predefined, i.e. the curricula of each class is usually known and fixed. Teachers are usually previously assigned to courses and the number of sessions of each course that the classes have to take is also known. The groups of students are not necessarily disjoint but in general most of them are. In a way, this simplifies the problem by helping to reduce the number of meetings. A simple formulation includes c classes, t teachers and p timeslots. A matrix M of size c x t holds the requirements, that is, each entry m(i,j) in the matrix indicates the number of times that the teacher j has to meet class i. The problem is to find: Xijk (i = 1,.., c; j = 1,.., t; k = 1,.., p) subject to p

∑ X ijk = mij

(5.1)

k =1 t

∑ X ijk

≤1

(5.2)

≤1

(5.3)

j =1

c

∑ X ijk i =1

where Xijk = 1 if class i and teacher j meet at period k and 0 otherwise. The required number of meetings between a teacher and a class must be scheduled (5.1). No teacher and no class can have more than one meeting at the same time (5.2 and 5.3 respectively). University course timetabling. This activity refers to the assignment of timeslots and locations so that meetings between lecturers and students can take place. Since students have more freedom for selecting their courses they are considered individually most of the times and therefore they are not pre-assigned to meetings. In addition, the creation of pairs lecturer-course is often part of the timetabling problem. The assignment of locations for the lectures is also an important part of the problem since the size and requirements of each group of students varies more than in the school timetabling problem. A simple formulation includes q courses where each course i consists of li sessions or lectures, there are p timeslots and r groups of courses S1,…Sr that have students in common. The problem is to find: Xik (i = 1,.., q; j = 1,…, r; k = 1,.., p) subject to p

∑ X ik

= li

(5.4)

∑ X ik

≤1

(5.5)

k =1

i∈S j

where Xik = 1 if a session (lecture) of course i is scheduled at period k and 0 otherwise. The required number of lectures of each course must be scheduled (5.4) and conflicting lectures must be scheduled at different times (5.5). University Examination Timetabling. This activity refers to the assignment of timeslots and locations so that students can take exams. As in university course timetabling, students are considered individually here but there may be a difference in the timeslots and rooms available. Usually, the number of available timeslots for the exams timetable is not the same as in the corresponding course timetable. Another important difference is that there is only one meeting to schedule for each course while in the previous problem one course can consist of several meetings. Also, the number and type of rooms available (seating capacity) and conflicts between exams provokes that some exams have to be scheduled in the same room or timeslot or apart of each other. Moreover, it is possible that one exam has to be split in two or more rooms. Assigning invigilators to the exams is sometimes part of the timetabling

activity here. A simple formulation include q exams, p timeslots, r groups of exams S1,…Sr that have students in common and gk is the maximum seating capacity at period k. The problem is to find:

Xik (i = 1,.., q; k = 1,.., p) subject to q

∑ X ik

≤ gk

(5.6)

∑ X ik

≤1

(5.7)

i =1

i∈S j

where Xik = 1 if exam i is scheduled at period k and 0 otherwise. There must be enough seating capacity for each exam (5.6) and conflicting exams must be scheduled at different times (5.7).

5.3 Feasibility and Timetable Quality The feasibility of solutions in the above timetabling problems varies according to the particular instance. In general, as specified in the above formulations, the hard constraints imply that no person (teacher, lecturer or student) can be present in two meetings at the same time, no location can be used for more than one meeting simultaneously and there are sufficient resources so that the meeting can take place (seating capacity, required equipment, etc.). However, in some cases these conditions are relaxed like in the case when clashes between lectures for the same student are unavoidable. Similarly, the criteria to measure the quality of the timetable differ from case to case. In general, in can be said that the ultimate goal when generating a timetable is to satisfy as much as possible the preferences of people (teachers, lecturers, students, administrators) and to utilise the available resources as efficiently as possible. Preferences include spread, compactness and balance of the timetable, free timeslots between meetings, meetings-free days, similarity with previous timetables, timetable flexibility, etc. Blakesley et.al. studied the problem of constructing academic timetables from a very different and interesting perspective: the student’s needs [Blakesly et.al. 1998]. They noted that constructing timetables that satisfy faculty and student preferences may have an unanticipated negative effect on the students needs since the availability of courses is reduced and the course completion time would be enlarged as a consequence. An example is the tendency to shorten the academic week, which increases the probability of conflicts. Sessions late on evening, too early in the mornings or on Fridays are commonly avoided and then the efficient utilisation of resources is affected. This tendency for preferring certain popular timeslots by faculty and students and avoiding other less popular timeslots has strong effect on the complexity of the problem. The number and variety of constraints (hard and soft) existing in academic timetabling problems makes almost impossible to list all of them. Some of the most common constraints reported in the literature are listed below. Unavailabilities and preassignments. Unavailabilities refer to the situation in which people or locations are not available during certain timeslots while preassignments refer to the situation in which certain meetings must be scheduled at the given time and/or location. Unavailabilities and preassignments are considered as hard constraints in most of the cases. Preference of locations and time. Sometimes there may be a list of preferred locations for scheduling a specific meeting or set of meetings. For example, when a lecture requires a room equipped with audiovisual aid. Similarly some timeslots may be preferred for certain meetings. For example, when exams with large number of students should be scheduled as early as possible in the timetable in order to have more time for marking. Restrictions on location and time. It may be that some locations and timeslots are not permitted for scheduling certain meetings. For example, when a music course cannot be scheduled in any room that is too close to a quiet area such as computer room or when an exam with large number of students cannot be scheduled in certain timeslots when many invigilators are not available.

Restriction on the sequence and grouping of meetings. These include all those constraints that refer to relationships between the meetings that limit the choices for finding suitable locations and times. Among the most common examples there are exams and lectures having people in common and imposed restrictions due to didactic or administrative reasons. For example, when two lectures of the same course cannot be scheduled in adjacent periods or when two difficult exams should be scheduled apart from each other. Coherence and balance of the timetable. A timetable should comply with certain requirements from the administrative and educational perspectives. For example, the minimum and maximum lectures per day or week that a lecturer can deliver should be according to the regulations. Another example is when the timetable should be balanced in terms of the number of lunch breaks and meetings-free timeslots. Location capacity. Any meeting can only take place in a location with the adequate seating capacity. This is usually a class of constraints that are strictly enforced, i.e. they are classified as hard or first order constraints. The most common situation is that the seating capacity cannot be exceeded but there are also cases in which assigning meetings to locations that are too big is not permitted. Length of the timetable. In the timetabling problems described in the literature it is often the case that the number of available timeslots is known and fixed. This is particularly true in the school and university course timetabling problems where the number of periods per day and per week is determined and cannot be exceeded. In these cases, the length of the timetable is an additional constraint that has to be taken into consideration. With respect to the examination timetabling problem, it is common than a constraint is imposed on the maximum length of the timetable although many times obtaining a shorter timetable is one of the goals in the problem.

5.4 Representation Schemes, Initialisation Strategies and Diversity Measures There are three aspects that deserve special attention in automated timetabling: the representation scheme used, the initialisation strategies to construct timetables and the metrics employed to measure similarity between timetables. It has been reported by researchers that the representation and initialisation strategies used, have an effect on the degree of difficulty to manage the constraints and on the performance of the searching algorithms [Burke et.al.1998][Corne&Ross1996][Corne&Ogden1998]. Also, measuring diversity of a set of timetables, which is crucial in Pareto optimisation, requires the design of specialised metrics according to the specific problem [Burke et.al.1998]. A variety of schemes have been used to represent timetables but in general there exist two classes of representations: direct and indirect also called explicit and implicit respectively. A direct representation encodes a timetable while an indirect representation encodes the steps to construct the timetable. For example, a direct representation of a timetable can be a string where each position represents an event (lecture, course, tutorial, exam) and the value is the timeslot in which the event is scheduled [Corne et.al.1994]. Another option is that each position in the string represents the timeslot and the value represents the event allocated to that timeslot. Graph models are also used to directly represent a timetable particularly when graph-colouring heuristics are used to tackle the problem [Carter et.al.1996]. It is also common that a matrix is used as a direct representation of the timetable [Schaerf1999a]. An example of indirect representation was used in [Corne&Ogden1998] for the automated generation of preaching timetables. That indirect representation consisted on a plan-building algorithm (constructive heuristic) that constructs the timetable step by step avoiding conflicts as it progresses. In their paper, Corne and Ogden also used a direct representation for their problem and compared three algorithms: hill-climbing, simulated annealing and evolutionary algorithms and found that the performance of each technique varied according to the representation scheme used. The hard constraints may or not be taken into account during the initialisation and searching processes and this will affect the definition of the solution space since infeasible solutions will be avoided or considered too. Similarly, soft constraints may or not be taken into account during the initialisation process but they are considered during the search. Then, initialisation techniques range from complete random creation of timetables without considering constraints (hard or soft) to very specialised constructive heuristics in which the meetings are scheduled according to their difficulty to be scheduled. The difficulty of each meeting to be scheduled has to be carefully assigned according to constraints in the particular problem instance. Corne and Ross experimented with various initialisation algorithms using different degrees of greediness called peckish initialisation [Corne&Ross1996]. This strategy resembles a greedy algorithm that occasionally makes mistakes. They expressed that a high proportion of highly fit initial solutions is not desirable but instead a high proportion of slightly fit initial solutions seems more preferable. In their

paper the authors presented evidence suggesting that peckish-uniform initialisation is beneficial across timetabling problems while greedy-feasibly strategies tend to be counterproductive. A study on various initialisation algorithms and their effect on the population diversity and on the performance of an evolutionary algorithm for the examination timetabling problem was presented in [Burke et.al.1998]. Each individual in the initial population was constructed using the process described next. For each exam to be scheduled a tournament of exams is built at random and one of the exams from the tournament is selected according to a given heuristic. Once the exam is chosen it is scheduled in the valid timeslot casing the least penalty breaking ties by preferring the earliest timeslot. Seven different heuristics were implemented for the tournament in that study, five of them were graph-colouring heuristics and the two others were a random sequencing strategy and the same random sequencing strategy followed by a hill-climbing process. Three diversity measures were used: “in same period”, “in same period as other exams” and “coverage of the search space” respectively. In the first method, two timetables are compared according to the number of occurrences were an exam is scheduled in the same timeslot in both solutions and this gives the similarity between the two timetables, then the average of this value over all pairs of timetables is used as a measure of the population diversity. In the second method, two timetables are compared by taking all pairs of exams that are scheduled in the same period in one of the timetables and if both exams are scheduled in the same period in the other timetable the number of occurrences is incremented. This process is repeated for each timeslot and the total number of occurrences is used as the similarity measure between the two timetables and the diversity of the whole population is calculated as in the previous method. These two methods measure the similarity directly between timetables while the third method used by the authors provides a measure of the diversity with respect to the whole solution space by counting the number of occurrences of the same pair exam-period that are present in the whole population and the average over all pairs exam-period is used as a measure of the population diversity. In their paper, the authors carried out various sets of experiments to investigate the effect of the different initialisation strategies on the quality and the diversity of the population produced by a memetic algorithm. They observed that the initialisation algorithm using the colour degree heuristic was particularly beneficial for the performance of their algorithm since the average quality of the population was very good without suffering too much reduction on the population diversity. Now the next section describes some approaches that have been proposed to tackle academic timetabling problems from a multiple criteria perspective.

6 Multiobjective Approaches for Academic Timetabling Although it is generally acknowledged that multiple criteria exist to evaluate solutions in academic timetabling problems, very few multiobjective metaheuristics have been applied to this class of problems. It has been pointed out that in the real world decision-makers prefer to have a selection of possible timetables from which to choose the most appropriate one [Carter1986]. However, the vast majority of approaches use a weighted sum of penalties for evaluating the fitness of solutions and only one timetable (the one with the lowest total penalty) is produced as a result. An automated algorithm will attempt to always obtain a lower penalty according to criteria defined in the algorithm but the workability of a timetable depends on how complete and realistic these criteria are. In practical problems there are three main reasons for imperfection of timetables: inaccurate prediction of student enrolment, mistakes in the events list or resources availability, and the nature of weights for the soft and hard constraints [Rankin 1996]. Some papers have reported on the application of strategies for producing various alternative solutions. For example, the combination of graph colouring techniques with heuristics was one of the first approaches that were used to produce several (not simultaneously) reasonable timetables [Burke et.al.1994]. In their approach, Burke et.al. use a weighted aggregating function to measure the quality of solutions but after one solution is generated, the decision-maker can redefine the selection criteria in order to obtain and review several solutions.

6.1 Multi-Phased Simulated Annealing An approach proposed to deal with the multiple criteria in timetabling problems is the use of multi-phased algorithms. For example, Thompson and Dowsland implemented a multi-phased simulated annealing algorithm for timetabling examinations [Thompson&Dowsland1996] [Thompson&Dowsland1996a]. The authors modelled the problem as a graph-colouring problem and the neighbourhood structure used was the change of colour in a single vertex that corresponds to moving an exam from one timeslot to another. In their approach, the first phase is used to tackle the first objective: the satisfaction of all the hard constraints while the second phase is used to optimise the

secondary objective: the minimisation of soft constraints violations. Since the decisions made in previous phases have an influence in the solutions that can be reached in later phases (the solution space may be disconnected), their multi-phased simulated annealing algorithm permits the alteration of decisions made in earlier phases as long as the quality of the solutions with respect to earlier objectives does not deteriorate. They experimented with various cooling schemes and investigated the relationship between the solution quality, the cooling schedule used and the execution time. They observed that whenever it is possible, slow cooling schedules should be used for simulated annealing in these problems and provided some recommendations on which cooling schedules to use according to the computation time available. In our opinion, the papers by Thompson and Dowsland are among the best reported studies on using simulated annealing for timetabling problems and among the few that approach these problems as multiple criteria optimisation problems. A similar study of the application of a multi-phased approach to examination timetabling and practical lab sessions timetabling was reported in [Dowsland1996]. In that investigation the author pointed out that the decision on which objectives or constraints are to be tackled in each phase depends not only on the importance of the objective but also on the difficulty to achieve it and on its relation with the neighbourhood structure defined (this has also noted by other researchers [Marett&Wright1996]). Ideally, when treating objectives in phases, one objective has to be tackled in each phase in order to eliminate the use of weights. However, this is not always possible in timetabling problems since the number of different objectives can be very large according to the number of constraints in the problem. Therefore, as in the multi-phased approaches described above, the constraints have to be grouped and each group is tackled in each stage of the algorithm. This causes the problem of still having to determine weights to reflect the relative importance of the constraints in the same group. Another drawback of multi-phased methods is that the solution obtained in an early phase requires to be fixed and this may lead to poor solutions in later phases since the solution space may be drastically reduced. A strategy to avoid this can be the implementation of backtracking mechanisms as proposed in [Thompson&Dowsland1996].

6.2 Multi-Phased Tabu Search Another multi-phased approach was described in [Alvarez-Valdes et.al.2000] for the course timetabling problem in Spanish universities. The first phase is an interactive process in which students select their courses and the second phase uses a tabu search algorithm for constructing the timetable. In the assignment phase the following criteria were used to measure the quality of timetables: students course selections must be respected, section enrolments should be balanced, sections maximum capacities must not be exceeded, clashes in students timetables should be avoided, students timetables should be as good as possible (measure in terms of number of lectures per day, number and length of holes in the timetable and moves between buildings), student language preferences should be respected. The first of the above criteria is the only hard constraint imposed since once the student has selected the course and the first phase of the system has confirmed that the enrolment is possible, it is a commitment between student and the university to respect the selection. The construction of the timetables is divided in two steps. In the first step, the best set of timetables is constructed for each student according to their selection of courses and without taking into consideration the balance of the course sections. In the second step a global timetable is constructed by combining the timetables of all students to obtain balanced section enrolments and minimise the decrease of the quality of each student timetable. A simple neighbourhood structure in which the section assignment of one student is changed was used and a neighbouring solution is then generated in three steps. First, a candidate list of students is generated with only those students that are involved in a violation of capacity constraint or all students if the solution is already feasible. Then one student is chosen at random from the candidate list. In the third step all possible exchanges for the selected student are evaluated and the best one is chosen provided is not a tabu move or if it is the aspiration criteria is satisfied, i.e. the exchange that produces the best improvement in the objective function. A strategic oscillation mechanism was implemented to vary the value of a weight for infeasibility used in the penalty function.

6.3 A Multiple Criteria Decision-making Technique Burke et.al. approached the examination timetabling problem specifically as a multiple criteria problem by dividing nine different constraints into three categories: room capacity, proximity of exams and time and order of exams [Burke et.al.2001] as shown below. Room capacities. One criterion was included in this group.




C1 represents the number of times that room capacities are exceeded.

Proximity of exams. Four criteria expressing the number of conflicts where exams are not adequately spread out in time were included here.





C2 represents the number of conflicts where students have exams in adjacent periods on the same day. C3 represents the number of conflicts where students have two or more exams in the same day. C4 represents the number of conflicts where students have exams in adjacent periods. C5 represents the number of conflicts where students have exams in overnight adjacent periods.

Time and order of exams. Also four criteria were included in this group. These criteria express the number of times that students are affected by inappropriate times of exams or by inappropriate order of exams.





C6 represents the number of times that student has an exam that is not scheduled in a time period of the proper duration. C7 represents the number of times that a student has an exam that is not scheduled in the required time period. C8 represents the number of times that a student has an exam that is not scheduled before/after another specified exam. C9 represents the number of times that a student has an exam that is not scheduled immediately before/after another specified exam.

According to Burke et.al., the above criteria are incommensurable and partially or totally conflicting at least in their problems. One hard constraint was considered in their paper: that conflicting exams must be scheduled in different timeslots. As in the multi-phased approaches described in the previous section, the approach by Burke et.al. also requires to set weights for expressing the relative importance of the different criteria within the same group. They used compromise programming, a technique from the multiple criteria decision-making field, as the basis for their solution method [Zeleny1973]. In compromise programming the strategy is to try finding compromise solutions which are close to the ideal point. An ideal point is defined in the criteria space as the vector containing the best possible value for each criterion. Their algorithm uses two phases: in the first one timetables of high quality are constructed using a graph-colouring heuristic and the second phase attempts to improve the timetables by using hillclimber and a heavy mutation operator. In each step of the preference space search, multiple applications of the hillclimber are followed by one application of the mutation operator until the distance between the solution and the ideal point has not decreased for a predefined number of iterations. One final solution is chosen from the set of obtained timetables, the one with the minimum distance from the ideal point. The authors noted in their experiments that the weights for each criterion and some parameters of the function to measure the distance from each solution to the ideal point, had a significant influence in the quality of the solutions obtained.

6.4 Multiobjective Evolutionary Algorithms One of the few applications reported in the literature of Pareto-based genetic algorithms to timetabling problems is the one by [Carrasco&Pato2001]. In that paper the authors tackled a bi-objective school timetabling problem with a modified version of the NSGA proposed in [Srinivas&Deb 1995]. The hard and soft constraints considered in their problem are listed below. Hard Constraints:






Each teacher and class is assigned to no more than one lesson and one room at a time period. Each class curriculum is respected. The daily bound of the number of lessons for each lesson cannot be exceeded. Rooms should be suitable for the lessons assigned to it. The teaching shift for each class must be respected.

Soft Constraints:




Each class and teacher should have a lunch break lasting at least one hour. The occurrence of gaps between teaching periods should be minimal for teachers and classes. Class and teacher preferences should be satisfied whenever possible.





The number of teacher and class shifts between teaching locations should be minimised. The number of days occupied with lessons should be minimised.

The two conflicting objectives were the minimisation of soft constraints violation from two competitive perspectives: teachers and classes. Penalties were assigned to the violation of constraints and the authors observed that the algorithm was very sensitive to the selection of these penalties. Since the NSGA uses fitness sharing, a measure of distance between two timetables xi and xj is required. Carrasco and Pato used the following metric for this purpose: L

d ( xi , x j ) =

∑ t (k , x i , x j )

k =1

L

(6.1)

where L is the total number of lessons and t(k,xi,xj) equals 1 if the lesson k occupies the same period in both solutions xi and xj and 0 otherwise. A direct representation was used in which a bi-dimensional matrix represents the timetable. Each row represents one room and each column represents one timeslot. Then, each cell in the matrix contains the lesson that will be taught in the given room at the given period. The creation of meetings (teacher-lesson-class) is done previously to the construction of the timetable. A constructive heuristic that starts scheduling the most difficult lessons (in terms of lesson duration and preferences of teachers and classes) first is used to initialise the population. An elitist secondary population composed of some of the non-dominated solutions from the main population was used. Specialised crossover and mutation operators were designed for the chromosome representation described above. The crossover operator was specially designed to create two offspring, one teacher-oriented and the other class-oriented. That is, their specific crossover operator attempts to produce elite timetables with respect to one of the objectives. A repair operator was employed for fixing the overlaps that are normally created by the crossover. The mutation operator consisted on removing a number of lessons from the timetable and then these lessons are re-scheduled in a way that the total penalisation is minimised. Although the authors mentioned in their paper that several experiments were carried out to assess the effect of the fitness sharing mechanism and the secondary population in the genetic algorithm, not detailed discussion of these effects was provided. However, they observed that the use of the secondary population was helpful and the algorithm found better timetables than those constructed manually. Another application of a Pareto-based genetic algorithm was reported in [Paquete&Fonseca2001] where the authors implemented a multiobjective evolutionary algorithm [Fonseca&Flemming1998] for the examination timetabling problem. In that paper, the authors used a direct chromosome encoding and a mutation operator with an independent mutation probability for each gene in the chromosome (no recombination operator was implemented). Each gene in the encoding represents an exam and the mutation probability for each gene is calculated according to the number of timeslots available for each exam and the degree of involvement of that exam in the violation of constraints. Their experiments sought to compare three aspects: Pareto ranking against linear ranking, independent mutation against single-position mutation and different levels of mutation bias. They reported that: the use Pareto ranking produced better performance in the algorithm, no difference was observed between the two mutation strategies and although difference was observed between groups of mutation rates, no more details were provided. One interesting aspect in the study by Paquete and Fonseca is that experiments were carried out considering the execution time as an additional objective and the independent mutation operator produced better performance in these experiments. Another interesting observation made by the authors was that each objective handling technique performed better in its own case, i.e. Pareto ranking provided better coverage of the objective space while linear aggregation was more effective in minimising the total number of constraint violations across the runs. This may represent an important clue for the implementation of non-dominated local search (mentioned in section 4.7) in timetabling problems. It can be noted that in fact, very few multiobjective metaheuristics have been applied to academic timetabling problems. However, many reports exist in the literature on the successful use of metaheuristics to the automated generation of academic timetables. An overview of the approaches (non-multiobjective) proposed for these problems is presented in the next section. The aim is to identify the main strategies that have contributed to the success of metaheuristics on single-objective academic timetabling problems and attempt to establish the basis for future research on multiobjective academic timetabling.

7 Review of Techniques for Academic Timetabling 7.1 Surveys on Academic Timetabling Several surveys on techniques and practical applications for academic timetabling problems are available in the literature. In 1985 De Werra described some formulations and models of some academic timetabling problems and also made a brief review of some of the heuristic methods applied to solve these problems [De Werra 1985]. Carter presented one of the earliest surveys on practical applications of examination timetabling in which a variety of graph colouring heuristics are described [Carter1986]. Several of the ideas that have supported the development of later algorithms are identified and described in that work and later developments were surveyed in [Carter&Laporte1996]. Later, Carter and Laporte presented another survey, this time on practical course timetabling considering the publications in the literature between 1980 and 1997 [Carter&Laporte1998]. Another survey on university timetabling is the one by [Burke et.al.1997] in which the authors described some practical issues about the timetabling process in academic institutions with special emphasis on British universities. Their survey also looked at some of the approaches that had been recently applied for university timetabling including genetic and memetic algorithms, simulated annealing, tabu search and constraints genetic programming. A recent survey on timetabling is the one by Schaerf that also provides references for other surveys in this field [Shaerf1999].

7.2 Methods Applied to Academic Timetabling In general, it can be said that among the solution approaches that have been applied to academic timetabling problems there are: heuristics simulating the human way of solving the problem, graph colouring techniques with and without backtracking, integer programming, network flow, simulated annealing, tabu search, evolutionary algorithms, constraint satisfaction, case-based reasoning, hyperheuristics, etc. The most successful methods so far appear to be a combination of metaheuristics with large amount of problem-specific knowledge. Graph colouring heuristics have been widely used to tackle timetabling problems not only as the basis for constructive heuristics but also as initialisation methods, local search heuristics and repairing methods.

7.2.1 Graph Colouring Heuristics An examination timetabling system based on graph colouring constructive heuristics and a backtracking mechanism was described in [Carter et.al.1994]. In that paper the exams were scheduled in non-decreasing order of the number of available periods remaining for each exam. A more detailed description of the algorithm implemented was later presented in [Carter et.al.1996]. The backtracking strategy consisted on the following procedure: when the next exam to be scheduled is in conflict with all periods, i.e. there is no available timeslot, some of the exams already scheduled are removed from the timetable. Some rules are defined in order to avoid cycling and limit the number of unscheduled exams during this backtracking process. In this second paper, the authors described their experiments using several variants of their approach. These variants were designed by using one of five graph-colouring heuristics, using backtracking or not, calculating cliques (minimum number of period required for the underlying graph colouring problem) or not and using costs for soft constraints or not. The sets of test problems in which the experiments were carried out, is one of the few timetabling data sets publicly available up to now. The authors found that the backtracking mechanism seemed to be quite beneficial in the performance of the heuristics particularly when the cost of soft constraints was considered.

7.2.2 Evolutionary Approaches A parallel genetic algorithm for the school timetabling problem was described in [Abramson&Abela1992]. The mating is carried out between periods independently which allows to preserve good packings of tuples from one timeslot to another. The crossover combines the packings between the same timeslots in two parents. Once the parents are mated the mutation is applied to the child with some probability. The crossover may produce infeasible timetables so a repair mechanism was implemented. The parallel implementation of the genetic algorithm was considered to take advantage of the concurrency when creating the new population since the mating operations are largely independent among them.

In 1994 Corne et.al. proposed a description for timetable problems attempting to include most of the academic timetabling problems [Corne et.al.1994]. This description was called the general examination/lecture timetabling problem (GELP) and consisted of a set of events (lectures, exams, seminars, practical sessions, etc.), a set of starting times, a set of locations where the events can take place and a set of agents that have a role to play in the events (lecturers, invigilators, classes, tutors, etc.). An assignment was defined as a 4-tuple (a,b,c,d) where event a starts a time b in location c and with agent d. In their evolutionary algorithm, the chromosome representation was a vector of length 3v where v is the number of events to schedule. Each group of three cells in the vector correspond to time, place and agent of the event i. A penalty function and an approximate evaluation strategy (delta evaluation) were used to make a faster evaluation of the candidate solution by considering only the recent changes made to the chromosome. Only the mutation operator was implemented consisting on a random selection of an event (gene) and then use of a tournament strategy to modify the allele for the gene according to the degree of constraints violation involving the chosen event. In each reproduction step an individual is generated as above then it replaces the less fit individual of the population only if the new individual is fitter. Some specialised recombinative operators based on sequential scheduling heuristics (including graph colouring) were proposed in [Burke et.al.1995] for the examination timetabling problem. Acting upon a direct representation of the problem these operators were designed specifically to maintain the feasibility of solutions. The common operation of these recombinative operators is as described below. Step 1. Two parents are selected to generate the child timetable. Step 2. For each time period and starting from the first one, Step 2.1. The operator first schedules in the child timetable those exams that are common in both parents in that period. Step 2.2. Then, the operator chooses other exams to be scheduled in the current period from those exams that are scheduled in one of the two parents or from those exams that were passed as unscheduled from the previous period. This selection of the next exam is made using a heuristic. An exam is scheduled in the current period only if no conflicts are created and if the seating capacity in that period is within the permitted limits. All exams that are scheduled in the current period in any of the parents and were not scheduled in the child timetable are passed as unscheduled to the next period. Burke et.al. investigated the application of random selection and various other heuristics for choosing the next exam to be scheduled in the Step 2.2 above. These included two graph-colouring heuristics and three other heuristics (and some combinations between them) specially designed using knowledge of the problem domain. Erben and Keppler employed specialised representation and genetic operators to maintain feasibility in their genetic algorithm for university course timetabling [Erben&Kepler1996]. They used a matrix with five dimensions to represent a chromosome where each gene represents the assignment of a class-teacher-lesson-room-period. Their initialisation procedure creates random feasible solutions without caring for soft constraints. Their mutation operator only makes changes between the events in a predefined time window (the set of events in the timetable within a set of given timeslots) using swaps, shifts and reassignments. The crossover operator is a very specialised one designed specifically for their chromosome representation. They noted that in many cases overnight runs were necessary in order to obtain satisfactory results. In their memetic algorithm for examination timetabling, Burke et.al. used mutation operators combined with an iterative improvement local search phase to deal only with feasible timetables [Burke et.al. 1996]. To represent a chromosome they used a list of genes where each gene is one timeslot that contains a sublist of rooms and each room contains the list of exams assigned to that room. In their approach, a light mutation operator chooses a number of exams and reschedules them maintaining feasibility. Then, the local search (hill-climbing) phase is applied followed by a heavy mutation operator that disrupts one or more periods (all exams in the period) in the timetable. Paetcher et.al. presented a memetic algorithm that used suggestion lists for the course timetabling problem [Paechter et.al.1996]. A suggestion list for each event is the list of feasible timeslots for allocation the event. Their genetic operators are specifically designed for being used with these suggestion lists. Three types of mutation are selected at random with predefined probabilities: blind mutation, selfish mutation and co-operative mutation. Blind mutation is the reinitialisation of the suggestion list. Selfish mutation steals the head of the list from one event to another. Co-operative mutation interchanges the heads between the lists of two events. A hybrid genetic algorithm for the school timetabling problem was presented in [Deris et.al.1999]. That approach consisted on applying a genetic algorithm to produce solutions by using genetic operators and then using constraintbased reasoning to evaluate the feasibility of the timetables and repair them when necessary. The chromosome

representation selected was a string in which each gene represents a meeting, and the alleles are the timeslot and room assigned to each meeting. This is a genetic algorithm that was maintained as general as possible and the simple genetic operators were used. The constraint-based reasoning is applied to all individuals after the genetic operators are used to produce the offspring in order to maintain the feasibility of solutions. The authors expressed that their genetic algorithm does not need to be modified on its components and the constraint-based reasoning reduces the search space and then speeds-up the search. The authors suggested that their approach has advantages over other techniques such as graph colouring techniques and other evolutionary approaches that use penalty functions to deal with constraints but did not compare their algorithm against these other methods. A multi-stage evolutionary algorithm that used problem decomposition was applied by Burke and Newall to the examination timetabling problem [Burke&Newall 1999]. Their approach consisted on applying a memetic algorithm but only to schedule a subset of the exams and then fixing these exams before applying the memetic algorithm to schedule the next subset of exams. In each phase of the multi-stage approach, the sets of exams to be scheduled are selected according to graph colouring heuristics. The exams already scheduled (fixed) for each period are merged to from “super events” so that in later stages the evaluation of the timetable can be done more quickly. The memetic algorithm used two mutation operators since the authors reported that previous experiments with recombination operators were not very successful. Burke and Newall carried out experiments to compare the performance of their hybrid algorithm against each of the individual components (sequencing heuristics and memetic algorithm) running on its own. They observed that the multi-stage evolutionary algorithm was capable of generating better timetables and reducing dramatically the computation time required for constructing the timetable. Ueda et.al. implemented a two-phase genetic algorithm for the course timetabling problem [Ueda et.al.2001]. In the first phase the classes are scheduled and in the second phase the rooms are assigned. Two direct representations were used, one representation in each stage of the algorithm and then, the two genotypes were combined to evaluate the timetable. Repairing heuristics were required to fix the timetables generated by the genetic operators. Erben investigated the application of the grouping genetic algorithm, proposed originally by [Falkenhauer1994], to the graph colouring problem and also to some examination timetabling problems [Erben2001]. A grouping genetic algorithm requires the design of very specialised chromosome representation and genetic operators and in his experiments Erben observed that a lot of effort was needed to tune the parameters of this algorithm. Recently, Socha et.al. applied an ant colony optimisation algorithm namely a max-min ant system to some test instances of the university course timetabling problem in which timeslots and rooms had to be assigned to lectures [Socha et.al.2002] . The search was limited to only feasible solutions. Each ant in the system walks along the list of events in the construction graph E x T (events and timeslots) and chooses a timeslot for each event at random using the pheromone matrix and heuristic information. This representation permitted the authors to use heuristics to sort the events according to their difficulty to be scheduled and this ordering is used by all ants when constructing the timetable. The approach was compared against a random restart local search using the same local search routine used in the ant system. The aim was to assess whether the ants learn or not to build timetables and the authors found in their experiments that the ant system produced better solutions that the random restart approach and in particular the later was unable to find feasible solutions for the large instances. Ross et.al. carried out a systematic study on the applicability of genetic algorithms to examination timetabling [Ross et.al.1998]. They used a direct representation of the timetable and weighted penalty functions to evaluate its quality. They observed that a direct representation may be inadequate for implementing a genetic algorithm for timetabling problems because the algorithm is not provided with co-ordination mechanisms on the effort to solve different parts of the problem. The authors expressed in their paper that it is often the case that the effect of crossover and mutation is not fully studied to determine their contribution to the solution of timetabling problems.

7.2.3 Simulated Annealing and Tabu Search Abramson approached the school timetable problem using simulated annealing and proposed sequential and parallel versions of this approach [Abramson1991]. A simple neighbourhood structure was used consisting on swapping the lectures between two time periods selected at random. A maximum number of successful and unsuccessful swaps is limited in each temperature level. Abramson experimented with different cooling rates and found that lower cooling rates produced better and more consistent solutions. In the parallel implementation, the proposal is to make several swaps at the same time provided these swaps are independent. Better speed-up of the parallel variant over the sequential one was reported for the larger problems. Dowsland presented case studies to show that simulated annealing and tabu search are suitable to design robust solutions for timetabling, rostering and scheduling problems and that the experiences from applying these techniques to one of this type of problems are frequently valuable to solve another type of new problems [Dowsland1998]. The

author also expressed that neighbourhood search methods are particularly attractive for these problems since natural neighbourhood structures exist. The tabu search implementation by Hertz is relatively simple but was one of the first approaches of this type reported in the literature for timetabling problems [Hertz1991]. Taking into account the availabilities of teachers and classrooms together with the required preassignments, a set of feasible timeslots is defined for each course to be scheduled. These lists of possible timeslots are computed only once at the beginning and they never change during the search. A simple neighbourhood structure was defined were a move consists on assigning a course section to a different timeslot but the moves considered are only those involving courses that are creating at least one conflict in the current timetable and infeasible solutions are part of the search space. Costa designed a very elaborated weighted penalty function to evaluate a timetable in a tabu search algorithm for the school timetabling problem [Costa1994]. During the search, when at least one of the hard constraints is violated only lectures contributing to these violations are selected to be moved. When no hard constraints are violated, only conflicting lectures are moved, i.e. lectures that contribute to the penalisation of soft constraints. A two-phases hybrid algorithm between tabu search and randomised non-ascendent local search was applied in [Schaerf1999a] to the school timetabling problem. Each method (tabu search and the randomised local search) acts on the solution in one of the two phases and using different neighbourhood structures. The randomised nonascendent method initialises the solution, makes more complex moves than the tabu search and shakes the timetable by walking through the plateau (by accepting candidate solutions with the same fitness as the current solution). The tabu search phase uses lists of variable size and adaptive relaxation of the hard constraints, i.e. infeasible timetables were also considered. White and Xie implemented a four-phase tabu search algorithm for the examination timetabling problem in which the two types of memory were used: recency-based (short term) and frequency-based (long term) [White&Xie2001]. The authors observed in their experiments that when both types of memory were used, the tenure of the short-term memory was not critical for the performance of the algorithm. Another strategy used in their approach was the relaxation of the tabu lists as an effective way of controlling the balance between diversification and intensification in the search. Several implementations of the tabu search algorithm for the examination timetabling problem were presented in [Di Caspero&Schaerf2001]. The authors used a randomised strategy to establish the tabu tenure: when a move is inserted in the tabu list, a random number indicating the number of iterations that the move will remain tabu is generated. All feasible and infeasible solutions are part of the solution space and the only constraints imposed at all times are the preassignments and unavailabilities. For helping to identify promising moves, lists that contain exams that are involved in violation of constraints are maintained. The authors experimented with several variations of the algorithm in order to evaluate the relative importance of the different components. Thompson and Dowsland presented a study in which they investigated the impact of the selection of various parameters in a simulated annealing algorithm when applied to a range of examination timetabling problems [Thompson&Dowsland1998]. In particular, the authors studied the effect of the cooling schedule and the neighbourhood structures on the performance of their algorithm. Three neighbourhood structures were compared. The first one was a standard structure where the timeslot for one single exam is changed. The second neighbourhood was based in Kempe chains and proposed specifically for graph colouring problems. The third neighbourhood structure used was a modification of the above called S-chains and the authors used the 3-chain in their experiments too. Preliminary experiments showed to the authors that 3-chain and Kempe chains neighbourhoods did not present clear difference and only the first two were used in the rest of their experimentation when comparing these structures on a variety of cooling schedules. All the schedules were geometric just using different values of cooling rates. The superiority of the Kempe chain neighbourhood over all the problems and cooling schedules was apparent. They concluded that the two-phases approach using Kempe chain neighbourhood and slow geometric schedule is a flexible and robust approach across the various problems. Further experiments showed that the way in which the neighbourhood is sampled also has an effect in the quality of the solution obtained.

7.2.4 Comparison Between Approaches It has been pointed out by several researchers that in general it is difficult to compare results on the performance of techniques when applied to academic timetabling problems due to the lack of widely accepted benchmark problems. Many papers on automated academic timetabling report results on the application of metaheuristics on specific real instances and many times the data sets are not available and when they are, the particularities of the problem or the

data format makes difficult their use by other researchers. Some of the few data sets publicly available are listed below.





Examination timetabling problems at the University of Toronto, ftp://ftp.mie.utoronto.ca/pub/carter/testprob. Examination timetabling problems at the University of Nottingham, ftp://ftp.cs.nott.ac.uk/ttp/Data. School timetabling problems in the OR Library, http://mscmga.ms.ic.ac.uk/jeb/orlib/tableinfo.html. University course timetabling problems, http://iridia.ulb.ac.be/~msampels/tt.data/.

Colorni et.al. addressed the high school timetable problem using three algorithms: simulated annealing, tabu search and genetic algorithms with and without local search [Colorni et.al.1998]. Their fitness function included infeasibilities as well as didactic and organisational costs. The simulated annealing algorithm used reheating and constraints relaxation (less restrictive fitness function) after reaching temperature zero. The tabu search algorithm also used constraints relaxation as a mechanism for diversification. The genetic algorithm used specialised operators and a filtering function (repairing heuristic) to deal with infeasibility. It was observed that when local search was not incorporated into the genetic algorithm, high costs for infeasibilities should be used in order to produce good results. On the other hand, they also observed that local search is well coupled with low infeasibility costs since if these costs are high, local search tries to recover feasibility by only affecting soft constraints. The explorative capability of the algorithm is improved by using low infeasibility costs because more infeasible timetables are accepted in the early iterations of the search. In general, Colorni et.al. observed in their experiments that simulated annealing performed the worst, being the tabu search the best closely followed by genetic algorithms coupled with local search. They highlighted that one advantage of the hybrid genetic algorithm over the other two approaches is that end-users have the flexibility of choosing within a set of different timetables (they did not referred to Pareto optimisation). Another comparison between various algorithms for course timetabling is available in [Saleh Elmohamed et.al.1998]. The authors compared a rule-based expert system and several variants of simulated annealing using different cooling schedules and reheating. Their best results were produced when the initial solutions are constructed with the rule-based preprocessor and then improved by the simulated annealing algorithms with adaptive cooling schedule and reheating. They found that the number of steps at each temperature level should be proportional to the neighbourhood size in order to maintain a high quality solution. The authors justified the selection of simulated annealing because it is relatively easy to implement and to combine with other heuristics but pointed out that among the drawbacks are: the need to carefully tune the algorithm, long execution times and several runs required.

7.2.5 Recent Proposed Techniques Even in recent papers related to timetabling, many approaches using single-solution local search methods are still widely used. For example Burke Newall studied the effect of parameters of local search on the quality of the solutions obtained when starting with good solutions on the examination timetabling problem [Burke&Newall2002]. Merlot et.al. implemented a three-stage algorithm consisting of a constraint programming phase to obtain a feasible timetable followed by a simulated annealing and a hill-climber for improvement of the solution for examination timetabling [Merlot et.al.2002]. A broker multi-agent algorithm that uses scheduling, unscheduling and rescheduling phases was proposed in [Lin2002]. In this algorithm, once the events are scheduled, those parts of the timetable that are the worst are unscheduled and a broker coordinates the work of various agents that attempt to rescheduled the unscheduled events. A proposed research direction has been the application of evolutionary algorithms to evolve the heuristics or algorithms to construct timetables instead of evolving the timetables directly [Ross et.al.1998]. Another research direction proposed recently is the investigation of methodologies for automatically and intelligently choosing an appropriate algorithm for the given problem [Burke et.al.2002a].

8 Multiobjective Approaches for Personnel Scheduling Personnel scheduling refers to the construction of shift patterns for employees and it is also known as rostering or employee timetabling. Personnel scheduling problems are multiple criteria problems that are similar to the academic timetabling problems because they also involve the construction of a schedule that satisfies as much as possible a number of diverse criteria. The criteria are usually also incommensurable and in conflict since they represent the interests of employees and employers and also working regulations. As in academic timetabling, few multiobjective

metaheuristics have been applied to personnel scheduling problems and this section attempts to describe in a brief manner, some of these approaches. No attempt is made to provide descriptions of personnel scheduling problems in this paper. Before the proposal of the Pareto simulated annealing for a range of multiobjective combinatorial optimisation problems in [Czyzak&Jaszkiewicz1998], that approach was applied to a multiobjective nurse scheduling problem in Polish hospitals [Jaszkiewicz1997]. In that paper, five objectives were identified, four minimisation objectives and one maximisation objective. One initial solution was generated by a constraint-based programming technique and multiple copies of this solution formed the initial population. Three types of neighbourhood structures were defined and in each iteration one of these was selected at random for generating the candidate solutions. Only feasible solutions were explored and if the chosen move violated any constraint another move was tried. The results were reported on a small test problem and the goal of producing better schedules that those manually generated was achieved. Mori and Tanaka applied a grouping genetic algorithm for timetabling conference programs that consists of first grouping the accepted papers into sessions according to the common topics and then allocating sessions to rooms and timeslots available [Mori&Tanaka2002]. Hard constraints are that no speaker can be in more that one session at a time and there is an upper limit in the number of papers per session. The soft constraints include that papers in the same session should have as many common keywords as possible, the difference between the number of papers per session and the capacity of the session should be minimised and the sessions in the same room should have as many keywords in common as possible. A similar multiple criteria approach to the one in [Burke et.al.2001] also using compromise programming was presented for the nurse scheduling problem in [Burke et.al.2002]. The main algorithm (based on tabu search) constructs a feasible schedule and iterative improvement of this initial schedule is tried by moving shifts between nurses and never accepting infeasible solutions. The ideal and anti-ideal points are estimated in order to make the mapping from the criteria space onto the preference space. Each personal schedule is considered separately and the sum of distances is used to measure the schedule fitness. El Moudani et.al. described a bi-criterion approach for the airlines crew rostering problem [El Moudani et.al.2001]. The airlines crew rostering problem refers to assigning crew staff to a set of pairings covering all the scheduled flights. A pairing is a sequence of flights that starts and ends at the same airline base while meeting all relevant legal regulations. A set of pairings is constructed in such a way that all the programmed flights are covered. Assigning crew staff to pairings should be done in such a way that the costs are minimised and the hard constraints are satisfied. Hard constraints include the regulations of Civil Aviation and the airlines internal agreements. Soft constraints of this problem include internal company rules, union agreements, office duties, holidays, assignment preferences and others. The nature of the soft constraints varies greatly between airlines. The authors tackled the airlines crew rostering problem from a bi-criterion perspective in which the first goal was to minimise airline operations cost and the second goal was to maximise the crew staff overall degree of satisfaction. The initial population of solutions was generated using a greedy heuristic specially designed to attempt the maximisation of the overall degree of satisfaction regardless of the operation costs. After this initial population is created, genetic operators are applied to generate new solutions with reduced operations cost at the expense of perhaps reduction on the degree of crew satisfaction. A direct chromosome representation was used in which each gene represents the pairing and the allele represents the crew member that has been assigned to that pairing. Three specific domain operators were implemented, crossover, mutation and inversion and a local search heuristic was designed to restrict the search space of the mutation and the inversion operators in order to speed-up the discovery of promising solutions. The authors reported that the application of the greedy heuristic to initialise the population required a very short computation time but in the subsequent application of the genetic operators, these operators did not show equivalent performances. They noted in their experiments that the crossover operator was very time consuming and it did not contribute too much to produce new promising solutions mainly due to the fact that the set of constrains to be checked is large. In the other hand, the mutation and inversion operators appeared to be more efficient in the generation of new promising solutions with relatively moderate computing times.

9 Other Relevant Research This section looks at some of the research work reported in the literature and that seems to be relevant for future research on multiobjective combinatorial optimisation in general and multiobjective scheduling problems in specific. Among the aspects that have been investigated there are the following: complexity of the landscape in multiobjective

combinatorial optimisation problems, effect of the evaluation method used to discriminate among solutions during the search on the performance of the algorithm and adapting operators during the search.

9.1 Complexity of the Landscape Wright and Marett carried out experiments to investigate the effect of the problem complexity on the performance of three algorithms: repeated descend, tabu search and simulated annealing in various sets of multiobjective combinatorial optimisation problems [Wright&Marett1996]. The aim of that study was to assess the performance of the three algorithms mentioned above with respect to the number of objectives and the shape of the local optima landscape. For studying the shape of the landscape, they measured the correlation between the sum of objectives’ improvement and the sum of objectives’ detriment when reaching local optima in a steepest descent run. When this correlation is close to +1, there is some conflict between the objectives. When the correlation is close to 0 the objectives are dissimilar or not affecting each other. When the correlation is close to –1 the objectives cooperate or reinforce each other. In their experiments they observed that the performance of the compared algorithms was dependent not only on the number of objectives but also on the correlation between the objectives measured as above. In multiobjective optimisation problems, some objectives may reinforce each other, conflict or be completely uncorrelated. Also, the improvement and detriment of each objective may be different and not constant during the search not only in their value but also in the frequency in which they change. These two aspects are related to the properties of the landscape and by studying them, an idea of the complexity of multiobjective combinatorial optimisation problems can be obtained.

9.2 Effect of the Evaluation Method Another important aspect that has been investigated by some researchers is the effect of the evaluation method used to discriminate among solutions during the search in Pareto optimisation. For example, Kokolo et.al. proposed the concept of α-dominance which is a relaxed dominance relation [Kokolo et.al.2001]. In α-dominance, a small detriment in one or more of the objectives is permitted if an attractive improvement in the other objective(s) is achieved. The hope is that by accepting α-dominating solutions, the search can be widened and the connectedness of the search space can be improved since α-dominating solutions may serve to reach to non-dominated solutions. Burke and Landa Silva applied this concept to the multiobjective optimisation of space allocation problems in academic institutions [Burke&LandaSilva2002]. They compared the performances of an evolutionary annealing algorithm and the PAES approach with respect to the form of dominance used and found that when using αdominance both algorithms obtained better non-dominated fronts. Additional experiments showed that this behaviour was not observed in the algorithms when the hard constraints in the test problems where treated as soft constraints, i.e. when the conditions of feasibility were relaxed so that it was easier to visit feasible solutions. Laumanns et.al. recently proposed the concept of ε-dominance (ε-dominance uses the same concept as the α-dominance proposed earlier by Kokolo et.al., i.e a relaxed from of dominance) to implement better archiving strategies that permit to overcome the difficulty of multiobjective evolutionary algorithms to converge towards the optimal Pareto front and maintaining a wide diversity in the population at the same time [Laumanns et.al.2002]. The use of subcost guided search was proposed by Wright to deal with compound-objective timetabling problems [Wright2001]. In that approach, an improvement of a subcost (objective) is preferred even if the overall cost or solution fitness is not improved at all or it is worsened. The hope is that the detriment suffered will be repaired later in the process since the improvement in one aspect of the solution (the subcost) permits to conduct a kind of guided diversification towards promising areas of the solution space. Wright carried out experiments with simulated annealing and threshold acceptance and found that the use of subcost guided search improved the performance of both algorithms.

9.3 Use of Adaptive Operators Applying different operators or heuristics at different stages of the search or according to the localisation of the solutions with respect to the Pareto optimal front may also be beneficial. For example, Salman et.al. proposed an approach based on a co-operative team of simple heuristics that generate non-dominated solutions for the multiple knapsack problem in a short computation time [Salman et.al.2002]. The team of heuristics co-operate in such a way that the solutions generated by one heuristic can be then improved by another one, or the adequate team of heuristics

can be formed to generate solutions for the given problem. Another option is to implement a set of local searchers that attempt to achieve self-improvement and ask for the help of other individuals in the population (perhaps by mating or sharing information) when they cannot achieve further improvement [Burke&LandaSilva2002b]. In this way, since the interactions between individuals are minimal and carried out in an asynchronous manner, the need for niching and fitness sharing strategies to maintain diversity is also reduced. In another example of co-operating heuristics, Cowling et.al. applied an approach that they call a hyperheuristic for solving a personnel scheduling problem [Cowling et.al.2001]. They describe a hyperheuristic as a strategy that is able to choose between a set of socalled low level heuristics based solely on performance indicators and not in the knowledge of the problem and then decide which heuristic to call at each moment during the search.

10 Final Remarks It is not within the scope of this paper to present a comprehensive review of multiobjective scheduling problems. It must be stressed that in particular for machine scheduling problems, there exist in the literature more complete studies on the multiobjective optimisation of these problems [Bagchi1999][[T'kindt&Billaut2002]. This tutorial has focused on the application of recent multiobjective metaheuristics for the optimisation of some types of multiobjective scheduling problems including machine scheduling, academic timetabling and personnel scheduling. The following remarks can be made after presenting this review:
The conditions of feasibility and the criteria used to measure the quality of solutions in multiobjective scheduling problems vary enormously between the different problem classes (machine scheduling, academic timetabling and personnel scheduling) and between particular instances. Machine scheduling problems (considering the singleobjective case too) are among the scheduling problems for which more benchmark theoretical models and test problems exist. Contrary to this, academic timetabling and personnel scheduling problems lack of widely accepted benchmark models and test problems. While in multiobjective machine scheduling problems several criteria have been clearly identified (makespan, tardiness, earliness, flow time, etc.), it is not clear which are the criteria that define the multiobjective nature of academic timetabling and personnel scheduling problems.
The application of modern multiobjective metaheuristics and in particular multiobjective evolutionary algorithms to multiple criteria scheduling problems is scarce. This is particularly true for academic timetabling and personnel scheduling which are real-world problems considered multiobjective by nature. Again, multiobjective machine scheduling problems (and in particular multiobjective flow shop scheduling) are among the multiple criteria scheduling problems for which more reports on the application of recent multiobjective metaheuristics exist in the literature. Within academic timetabling problems, the variant that has received more attention from the multiobjective perspective appears to be examination timetabling.
There exist several common strategies that can be identified in the reported applications of metaheuristics for the multiobjective optimisation of the multiple criteria scheduling problems considered in this paper. The importance of local search and problem domain knowledge is evident for the successful implementation of multiobjective metaheuristics to tackle multiple criteria scheduling problems. Even in recent approaches, local search continues to play an important role particularly when tackling academic timetabling and personnel scheduling problems. Weighted aggregating functions are still widely used in many approaches (in particular those using some form of local search) proposed for multiple criteria scheduling problems and they appear to be adequate when problems are highly constrained. Graph colouring heuristics are widely used in methods for academic timetabling problems and they can be considered to be implemented as operators in more elaborated approaches. In academic timetabling and personnel scheduling problems, due to the existence of lots of hard constraints, accepting infeasible solutions during the search has been helpful to improve the connectedness of the solution space and to widen the search in a number of applications. Also, the use of elite solutions according to each of the existing criteria has been successfully used in some recent approaches for multiobjective scheduling problems.
In the few reported implementations of multiobjective evolutionary algorithms to multiple criteria scheduling problems, several components such as representation schemes, initialisation strategies, genetic operators and repairing mechanisms need to be specially designed using problem domain knowledge. Also, in many approaches only mutation operators are used in evolutionary algorithms for scheduling problems (including the singleobjective case too) and sometimes these operators are in fact very elaborated heuristics designed to bias the search towards promising regions. Given the applications of multiobjective metaheuristics to multiple criteria scheduling problems reported in the literature and the related research that has been summarised in section 9 above, some research directions that could be explored towards more research in this area are proposed below.


Exploit the experiences obtained from research in some of these problems (such as multiobjective flow shop) to produce successful approaches in other multiobjective scheduling problems. For example, the use of weighted vectors to specify search directions towards the Pareto optimal set, the tuning of local search, the selection of adequate genetic operators, the balance between local search and genetic search in hybrid approaches, the use of elitist strategies, the study of the impact of selection mechanisms and diversity measures on the performance of the algorithm, the adaptation of genetic operators probabilities during the search, the use of exact methods or heuristics to quickly approximate the Pareto optimal front followed by heuristics designed to improve the quality of this approximation, etc.
Given the importance of local search in this context, it would be interesting to put more effort on studying the landscape in multiobjective scheduling problems in order to assess the complexity of these problems and design better local search components. Also, some strategies that have been proposed to improve local search for singleobjective combinatorial optimisation could also be incorporated into multiobjective metaheuristics. For example, the use of various neighbourhood structures or teams of local search heuristics according to the objective being optimised could be useful. Another example is changing the fitness landscape for improving the connectedness of the solutions space. Then using different evaluation methods to discriminate solutions during the search (aggregating functions and other forms of dominance) can be helpful in order to obtain better results for multiobjective scheduling problems.
More application of multiobjective metaheuristics to multiple criteria scheduling problems could be encouraged by either applying modern multiobjective metaheuristics after incorporating those operators and heuristics that have been successful in previous single-solution approaches for these problems or by extending single-objective metaheuristics for these problems towards multiobjective variants by incorporating some concepts and strategies from modern multiobjective metaheuristics.
Multiple criteria academic timetabling and personnel scheduling problems should be investigated in order to identify the criteria that should be considered when tackling these problems from a multiobjective perspective and also investigate the conflicting and incommensurable nature of these criteria.

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